MATHEMATICAL    WRINKLES 

FOR  TEACHEES  AND  PRIVATE  LEARNERS 
coN&idwirtar  OF  •       ••*  ' 


KNOTTY  PROBLEMS;    MATHEMATICAL   RECREATIONS 
ANSWERS  AND  SOLUTIONS;  RULES  OF  MENSURA- 
TION; SHORT  METHODS;   HELPS,  TABLES,  ETC. 


BY 


SAMUEL   I.    JONES 

PR0FSS80B   OF   MATHEMATICS   IK   THE   OUNTBR   BIBLICAL 
AND    LITEBABT    COLLEGE,    OUNTEB,    TEXAS 


PRICE  $1.65,  NET 


PUBLISHED  BY 

SAMUEL   L    JONES 

GUNTER,  TEXAS 


^  fA  I  ti 


V\*\f 


()  f:  ^    i  ?  '  ^'^■"^f^-'^-^^^-^^ 


Copyright,  1912, 
By  SAMUEL  I.  JONES. 

All  rights  reserved. 


NcriDoolr  i^reaa 

J.  8.  Cushing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Masa.,  U.S.A. 


"  Albert  Smith,  in  one  of  his  amusing  novels,  describes 
a  woman  who  was  convinced  that  she  suffered  from  '  cob- 
wigs  on  the  brain.*  This  may  be  a  very  rare  complaint, 
but  in  a  more  metaphorical  sense,  many  of  us  are  very 
apt  to  suffer  from  mental  cobwebs,  and  there  is  nothing 
equal  to  the  solving  of  puzzles  and  problems  for  sweeping 
them  away.  They  keep  the  brain  alert,  stimulate  the 
imagination,  and  develop  the  reasoning  faculties.  And  not 
only  are  they  useful  in  this  indirect  way,  but  they  often 
directly  help  us  by  teaching  us  some  little  tricks  and 
'wrinkles*  that  can  be  applied  in  the  affairs  of  life  at 
the  most  unexpected  times,  and   in   the   most   unexpected 

wavs." 

H.  E.  DUDENEY. 


Ill 


«00773 


CONTENTS 

CHAPTBB  PAOB 

I.     Arithmetical  Problems 1 

II.  Algebuaic  Problems   ........  25 

III.  Geometrical  Exercises 33 

IV.  Miscellaneods  Problems 48 

V.     Mathematical  Recreations 68 

VL     ExAMiNATiox  Questions 113 

Vn.     Answers  and  Solutions 163 

VIII.     Short  Methods 228 

IX.     Quotations  on  Mathematics 245 

X.     Mensuration 258 

XI.     Miscellaneous  Helps 285 

Xn.     Tables 304 


PREFACE 

The  following  pages  contain  many  mathematical  problems, 
puzzles,  and  amusements  of  past  and  present  times.  They 
have  a  long  and  interesting  history  and  are  part  of  the  inher- 
itance of  the  school. 

This  book  is  intended  to  be  a  helpful  companion  to  teachers, 
and  to  impart  to  students  a  knowledge  of  the  application  of 
mathematical  principles,  which  cannot  be  obtained  from  text- 
books. 

The  present-day  teacher  has  little  time  for  the  selection  of 
suitable  problems  for  supplementary  work.  This  book  is  de- 
signed to  meet  the  requirements  of  teachers  who  feel  such 
extra  assignments  essential  to  thorough  work.  Whatever  text 
is  used,  the  necessity  for  a  work  of  this  kind  is  felt  from  the 
fact  that  fresh  problems  produce  interest  and  stimulate  inves- 
tigation. 

Originality  is  not  claimed  for  all  of  the  problems,  but  for 
many  of  them.  They  have  been  compiled  from  various  sources. 
The  author's  aim  has  been  to  select  problems  not  only  instruc- 
tive, but  also  interesting  and  amusing. 

The  rules  of  Mensuration  and  Short  Methods  have  been 
included  because  of  their  usefulness.  On  account  of  the  vari- 
ous helps  placed  in  this  book,  it  will  serve  as  a  handbook  of 
mathematics  to  both  teachers  and  pupils. 

The  solutions  to  only  part  of  the  problems  are  given.  In 
some  cases  solutions  of  considerable  length  are  given,  but  at 
other  times  only  the  answers  are  given.  Had  the  full  solu- 
tions and  proofs  been  given  in  every  case,  either  half  the  prob- 
lems would  have  had  to  be  omitted,  or  the  size  of  the  book 
greatly  increased. 

vU 


viii  PREFACE 

The  author  acknowledges  his  indebtedness  to  many  friends 
for  helpful  suggestions.  Specially  is  he  under  obligation  to  the 
late  Dr.  G-.  B.  M.  Zerr,  Philadelphia,  Pa.,  for  critically  reading 
the  manuscript.  A  few  of  his  solutions  published  in  the  lead- 
ing Mathematical  Journals  have  been  used  on  account  of  their 
beauty  and  simplicity.  He  is  indebted  to  Dr.  H.  Y.  Benedict 
and  Mr.  J.  W.  Calhoun,  of  the  University  of  Texas,  for  read- 
ing the  manuscript  and  offering  many  valuable  suggestions 
and  criticisms.  He  is  very  thankful  to  Dr.  George  Bruce  Hal- 
sted,  head  of  the  department  of  mathematics  of  the  Colorado 
State  Teachers'  College  at  Greeley,  for  criticising  the  Defini- 
tions, Historical  Notes,  and  Classifications.  He  is  also  specially 
indebted  to  Professor  Dow  Martin,  of  the  Biblical  and  Lit- 
erary College  of  Gunter,  Texas,  for  reading  and  correcting  the 
proof-sheets. 

Any  correction  or  suggestion  relating  to  these  problems  and 
solutions  will  be  most  thankfully  received. 

It  is  hoped  that  this  small  volume  may  produce  higher  and 
more  noble  results  in  awakening  a  real  love  and  interest  among 
the  great  body  of  teachers  and  students  for  the  study  of  math- 
ematics, "the  oldest  and  the  noblest,  the  grandest  and  the 
most  profound,  of  all  sciences." 

SAMUEL   I.   JONES. 

GuNTEB, .  Texas. 


MATHEMATICAL  WRINKLES 


ARITHMETICAL  PROBLEMS 

1.*  Between  3  and  4  o'clock  I  looked  at  my  watch  and  noticed 
the  minute  hand  between  5  and  6 ;  within  two  hours  I  looked 
again  and  found  that  the  hour  and  minute  hands  had  exchanged 
places.     What  time  was  it  when  I  looked  the  second  time  ? 

2.*  A  tree  120  feet  high  was  broken  in  a  storm,  so  that  the  top 
struck  the  ground  40  feet  from  the  foot  of  the  tree.  How  long 
was  tlie  part  of  the  tree  that  was  broken  over  ? 

3.  How  many  acres  does  a  square  tract  of  land  contain,  which 
sells  for  $80  an  acre,  and  is  paid  for  by  the  number  of  silver 
dollars  that  will  lie  upon  its  boundary  ? 

4.*  The  area  of  a  rectangular  field  is  30  acres,  and  its  diag- 
onal is  100  rods.     Find  its  length  and  breadth. 

5.*  Suppose  two  candles,  one  of  which  will  burn  in  4  hours 
and  the  other  in  6  hours,  are  lighted  at  once.  How  soon  will 
one  be  four  times  the  length  of  the  other  ? 

6.*  While  a  log  2  feet  in  circumference  and  10  feet  long 
rolls  200  feet  down  a  mountain  side,  a  lizard  on  the  top  of  the 
log  goes  from  one  end  to  the  other,  always  remaining  on  top. 
How  far  did  the  lizard  move  ? 

7.  How  many  calves  at  $3.50,  sheep  at  $1.50,  and  lambs  at 
$  .60  per  head,  can  be  bought  for  $  100,  the  total  number  bought 
being  100  ? 

•  Problems  denoted  by  (•)  are  algebraic  or  geometrical.  They  are  placed 
here  because  arithmetical  solutions  are  often  demanded. 

1 


2  mathemat,ica;l  wrinkles 

8.  A, mar.  yfills  to  his-  wife-i  of. his  estate,  and  the  remain- 
ing I  to  ^iS'SOrr,  if'SL4ch  siie^ji'ld"  be  ^born ;  but  |  of  it  to  the  wife 
and  the  other  i  to  the  daughter,  if  such  should  be  born.  After 
his  death  twins  are  born,  a  son  and  a  daughter.  How  should 
the  estate  be  divided  so  as  to  satisfy  the  will  ? 

9.  What  is  the  value  of  4^^ ,  when  n  =  0? 

10.  A  room  is  30  feet  long,  12  feet  wide,  and  12  feet  high. 
On  the  middle  line  of  one  of  the  smaller  side  Avails  and  1  foot 
from  the  ceiling  is  a  spider.  On  the  middle  line  of  the  oppo- 
site wall  and  11  feet  from  the  ceiling  is  a  fly.  The  fly  being 
paralyzed  by  fear  remains  still  until  the  spider  catches  it 
by  crawling  the  shortest  route.  How  far  did  the  spider 
crawl  ? 

11.  I  found  $10;  what  was  my  gain  per  cent? 

12.*  A  conical  glass  is  4  inches  high  and  6  inches  across 
at  the  top.  A  marble  is  within  the  glass,  and  water  is  poured 
in  till  the  marble  is  just  immersed.  If  the  amount  of  water 
poured  in  is  ^  the  contents  of  the  glass,  what  is  the  diameter 
of  the  marble  ? 

13.  A  banker  discounts  a  note  at  9  %  per  annum,  thereby 
getting  10  %  per  annum  interest.  How  long  does  the  note 
run  ? 

14.  In  extracting  the  square  root  of  a  perfect  power  the 
last  complete  dividend  was  found  to  be  1225.  What  was  the 
power  ? 

15.*  Mr.  Smith  has  a  lawn  the  dimensions  of  which  are  to 
each  other  as  3  to  2.  If  he  should  increase  each  dimension  one 
foot,  the  lawn  would  cover  651  square  feet  of  land.  What  are 
the  dimensions  of  the  lawn  ? 

16.  A  merchant  marked  his  goods  to  gain  80  %,  but  on  ac- 
count of  using  an  incorrect  yardstick,  gained  only  40  %. 
Find  the  length  of  the  measure. 


ARtTHMETICAL  PROBLEMS  3 

17.*  The  area  of  a  triangle  is  24,276  square  feet,  and  its 
sides  are  in  proportion  to  the  numbers  13,  14,  and  15.  Find 
the  length  of  each  side. 

18.  Between  2  and  3  o'clock,  I  mistook  the  minute  hand 
for  the  hour  hand,  and  consequently  thought  the  time  55  min- 
utes earlier  than  it  was.     What  was  the  correct  time  ? 

19.  A  slate  including  the  frame  is  9  inches  wide  and  12 
inches  long.  The  area  of  the  frame  is  \  of  the  whole  area,  or 
J  of  the  area  inside  the  frame.  What  is  the  width  of  the 
frame  ? 

20.  If  6  acres  of  grass,  together  with  what  grows  on  the 
6  acres  during  the  time  of  grazing,  keep  16  oxen  12  weeks,  and 
9  acres  keep  26  oxen  9  weeks,  how  many  oxen  will  15  acres 
keep  10  weeks,  the  grass  growing  uniformly  all  the  time  ? 

21.  A  boy  on  a  sled  at  the  top  of  a  hill  200  feet  long,  slides 
down  and  runs  half  as  far  up  another  hill.  He  sways  back 
and  forth,  each  time  going  ^  as  far  as  he  came.  How  far  will 
he  have  traveled  by  the  time  he  comes  to  a  halt  ? 

22.  3  +  3x3-3-3-3=? 

23.  2h-2--2--2--2x2x  2x2-i-0x  2=? 

24.  3^3-^3^3x3x3x0x  3=  ? 

25.  A  fly  can  crawl  around  the  base  of  a  cubical  block  in 
4  minutes.  How  long  will  it  take  it  to  crawl  from  a  lower 
corner  to  the  opposite  upper  corner? 

26.  A  squirrel  goes  spirally  up  a  cylindrical  post,  making  a 
circuit  in  each  4  feet.  How  many  feet  does  it  travel  if  the 
post  is  16  feet  high  and  3  feet  in  circumference  ? 

27.  If  the  cloth  for  a  suit  of  clothes  for  a  man  weighing  216 
pounds  costs  $  16,  what  will  be  the  cost  of  enough  cloth  of  the 
same  quality  for  a  man  of  similar  form  weighing  512  pounds  ? 

28.  A  ball  12  feet  in  diameter  when  placed  in  a  cubic  room 
touches  the  floor,  ceiling,  and  walls.     What  must  be  the  diam- 


4  MATHEMATICAL  WEINKLES 

eter  of  8  smaller  balls,  which,  will  touch  this  ball  and  the  faces 
of  the  given  cube  ? 

29.  At  what  time  between  3  and  4  o'clock  is  the  minute 
hand  the  same  distance  from  8  as  the  hour  hand  is  from  12  ? 

30.*  By  cutting  from  a  cubical  block  enough  to  make  each 
dimension  2  inches  shorter  it  is  found  that  its  solidity  has 
been  decreased  39,368  cubic  inches.  Find  a  side  of  the  original 
cube. 

31.  A  number  increased  by  its  cube  is  592,788.  Find  the 
number. 

32.*  The  difterence  of  two  numbers  is  40 ;  the  difference  of 
their  squares  is  4800.     What  are  the  numbers  ? 

33.  A  man  can  row  upstream  in  3  hours  and  back  again  in 
2  hours.  Determine  the  distance,  the  rate  of  the  current  being 
1  mile  per  hour. 

34.  A  rented  a  farm  from  B,  agreeing  to  give  B  i  of  all  the 
produce.  During  the  year  A  used  90  bushels  of  the  corn 
raised,  and  at  settlement  first  gave  B  20  bushels  to  balance  the 
90  bushels  and  then  divided  the  remainder  as  if  neither  had 
received  any.     How  much  did  B  lose  ?  x 

35.  A  certain  number  increased  by  its  square  is  equal  to 
13,340.     Find  the  number. 

36.*  The  cube  root  of  a  certain  number  is  10  times  the 
fourth  root.     Find  the  number. 

37.  A  number  divided  by  one  more  than  itself  gives  a 
quotient  yL.     What  is  the  number  ? 

38.  What  do  I  pay  for  goods  sold  at  a  discount  of  50,  25, 
and  100  %  off,  the  list  price  being  $  50  ? 

39.  If  an  article  had  cost  ^  less,  the  rate  of  loss  would  have 
been  ^  less.     Find  the  rate  of  loss. 

40.  A  merchant  having  been  asked  for  his  lowest  prices  on 
shoes,  replied,  "  I  give  a  certain  per  cent  off  for  cash,  the  same 


ARITHMETICAL   PROBLEMS  5 

per  cent  off  the  cash  price  tu  ministers,  and  the  same  per  cent 
off  the  price  to  ministers  to  widows."  The  price  to  widows  is 
l^f  of  the  marked  price.  What  per  cent  does  he  give  off  for 
cash? 

41.  If  James  had  $40  more  money  he  could  buy  20  acres  of 
land,  or  with  $  80  less  he  could  buy  only  10  acres.  How  much 
money  has  he  and  what  is  the  value  of  an  acre  ? 

42.  What  is  the  least  number  of  gallons  of  wine,  expressed 
by  a  whole  number,  that  will  exactly  fill,  without  waste,  bottles 
containing  either  j,  |,  ^,  or  |  gallons  ? 

43.  I  sold  a  house  and  gained  a  certain  per  cent  on  my  in- 
vestment. Had  it  cost  me  20  %  less,  I  should  have  gained  30  % 
more.     What  per  cent  did  I  gain  ? 

44.  Goods  marked  to  be  sold  at  50  and  10  %  discount  were 
disposed  of  by  an  ignorant  salesman  at  60  %  from  the  list  price. 
What  was  the  loss  on  cash  sales  amounting  to  S  15,000? 

45.  I  paid  $  10  cash  for  a  bill  of  goods.  What  was  the  list 
price,  if  I  received  a  discount  of  50,  25,  20,  and  10  %  off  ? 

46.  My  clock  gains  10  minutes  an  hour.  It  is  right  at 
4  P.M.  What  is  the  correct  time  when  the  clock  shows  mid- 
night of  the  same  day  ? 

47.  Two  men  working  together  can  saw  5  cords  of  wood  per 
day,  or  they  can  split  8  cords  of  wood  when  sawed.  How 
many  cords  must  they  saw  that  they  may  be  occupied  the  rest 
of  the  day  in  splitting  it  ? 

48.  A  grocery  merchant  sells  goods  at  80  %  profit  and  takes 
eggs  in  trade  at  market  price.  If  2  eggs  in  each  dozen  are 
bad,  find  his  per  cent  gain. 

49.  A  hollow  sphere  whose  diameter  is  6  inches  weighs  |  as 
much  as  a  solid  sphere  of  the  same  material  and  diameter. 
How  thick  is  the  shell  ? 

50.  If  a  bin  will  hold  20  bushels  of  wheat,  how  many 
bushels  of  apples  will  it  hold  ? 


6  MATHEMATICAL   WRmKLES 

51.  What  per  cent  in  advance  of  the  cost  must  a  merchant 
mark  his  goods  so  that  after  allowing  5  %  of  his  sales  for  bad 
debts,  and  an  average  credit  of  6  months,  and  7  %  of  the  cost 
of  the  goods  for  his  expenses,  he  may  make  a  clear  gain  of 
12-1-  ^  Qf  lY^Q  f^^.g^  cQs^  Qf  ^\^Q  goods,  money  being  worth  6  %  ? 

52.  A  teacher  in  giving  out  the  dividend  84,245,000  was  mis- 
understood by  his  pupils,  who  reversed  the  order  of  the  figures 
in  millions  period.  The  quotient  obtained  was  36,000  too 
small.     What  was  the  divisor  ? 

53.  Three  men  bought  a  grindstone  20  inches  in  diameter. 
How  much  of  the  diameter  must  each  grind  off  so  as  to  share 
the  stone  equally,  making  an  allowance  of  4  inches  waste  for 
the  aperture  ? 

54.  James  is  30  years  old  and  John  is  3  years  old.  In  how 
many  years  will  James  be  5  times  as  old  as  John  ? 

55.  A  merchant  sold  a  piano  at  a  gain  of  40  %.  Had  it  cost 
him  $400  more,  he  would  have  lost  40  %.  What  did  it  cost 
him? 

56.  A  steamer  goes  20  miles  an  hour  dow^nstream,  and  15 
miles  an  hour  upstream.  If  it  is  5  hours  longer  in  coming  up 
than  in  going  down,  how  far  did  it  go  ? 

57.  A  and  B  together  can  do  a  piece  of  work  in  24  days. 
If  A  can  do  only  |  as  much  as  B,  how  long  will  it  take  each 
of  them  to  do  the  work  ? 

58.  The  sum  of  two  numbers  is  80 ;  the  difference  of  their 
squares  is  1600.     What  are  the  numbers  ? 

59.  When  a  man  sells  goods  at  a  price  from  which  he  re- 
ceived a  discount  of  33  J-  %,  what  is  his  gain  per  cent  ? 

60.  6-6--6  +  6x2-2=? 

61.  3--3--3^3--3--i--i--i--i=? 

62.  How  much  water  will  dilute  5  gallons  of  alcohol  90  % 
strong  to  30  %  ? 


ARITHMETICAL   PROBLEMS    '  7 

63.  I  bought  a  house  and  lot  for  $  1000,  to  be  paid  for  in  5 
equal  payments,  interest  at  10%,  payable  annually  ;  payments 
to  be  cash,  1,  2,  3,  and  4  years  from  date  of  purchase.  What 
was  the  amount  of  each  payment  ? 

64.  I  buy  United  States  4%  bonds  at  106,  and  sell  them 
in  10  years  at  102.     What  is  my  rate  of  income  ? 

65.  If  a  melon  20  inches  in  diameter  is  worth  20  cents, 
what  is  one  30  inches  in  diameter  worth  ? 

66.  The  difference  between  the  true  discount  and  the  bank 
discount  of  a  note  due  in  90  days  at  6  %,  is  $.90.  What  is 
the  face  of  the  note  ? 

67.  A  writing  desk  cost  a  merchant  $  20.  At  what  price 
must  it  be  marked  so  that  the  marked  price  may  be  reduced 
40  %  and  still  50  %  be  gained  ? 

68.  A  man  agreed  to  work  12  days  for  $  18  and  his  board, 
but  he  was  to  pay  $1  a  day  for  his  board  for  every  day  he 
was  idle.  He  received  $8  for  his  work.  How  many  days 
did  he  work  ? 

69.  A  druggist,  by  selling  10  pounds  of  sulphur  for  a  certain 
sum,  gained  50%.  If  the  cost  of  sulphur  advances  20%  in 
the  wholesale  mai-ket,  what  per  cent  will  the  druggist  now 
gain  by  selling  7^  pounds  for  the  same  sum  ? 

70.*  The  head  of  a  fish  is  9  inches  long.  The  tail  is  as  long 
as  the  head  and  .V  of  the  body,  and  the  body  is  as  long  as  the 
head  and  tail.     What  is  the  length  of  the  fish  ? 

71.  In  a  corner  of  a  bin  I  pour  some  grain  which  extends 
up  the  wall  8  feet,  and  whose  base  is  measured  by  a  circular 
line  10  feet  distant  from  the  corner.  How  many  bushels  in 
the  pile  ? 

72.  A  substance  is  weighed  from  both  arms  of  a  false  bal- 
ance, and  its  apparent  weights  are  4  pounds  and  16  pounds. 
Find  its  true  weight. 


8  MATHEMATICAL   WEINKLES 

73.  When  wheat  is  worth  $.90  a  bushel,  a  baker's  loaf 
weighs  9  ounces.  How  many  ounces  should  it  weigh  when 
wheat  is  worth  $  .72  a  bushel  ? 

74.  The  difference  between  the  interest  of  $  700  and  $  300 
for  the  same  time  at  6  %  is  $  84.     Find  the  time. 

75.  What  is  the  price  of  10  %  stocks  that  yield  a  profit 
equal  to  that  of  5  %  bonds  bought  at  80  ? 

76.  If  I  sell  oranges  at  8  cents  a  dozen,  I  lose  30  cents ;  but 
if  I  sell  them  at  10  cents  a  dozen,  I  gain  12  cents.  How 
many  have  I,  and  what  did  they  cost  me  ? 

77.  If  a  man  can  swim  across  a  circular  lake  in  20  minutes, 
how  long  will  it  take  him  to  ride  twice  around  it  at  twice  his 
former  rate  ? 

78.  If  f  of  the  time  past  noon,  plus  4  hours,  equals  f  of  the 
time  to  midnight  plus  3  hours,  what  is  the  time  ? 

79.  A  horse  steps  more  than  30  and  less  than  50  inches  at 
each  step.  If  he  takes  an  exact  number  of  steps  in  walking 
259  inches  and  an  exact  number  in  walking  407  inches,  what 
is  the  length  of  his  step  ? 

80.  I  sold  two  horses  for  $  200.  I  gained  10  %  on  the  first 
and  20  %  on  the  second.  How  much  did  each  cost  if  the  sec- 
ond cost  $  20  more  than  the  first  ? 

81.  A  thief  is  27  steps  ahead  of  an  officer,  and  takes  8  steps 
while  the  officer  takes  5 ;  but  2  of  the  officer's  steps  are  equal  to 
5  of  the  thief's.     In  how  many  steps  can  the  officer  catch  him  ? 

82.  A  tree  is  60  feet  high,  which  is  f  of  f  of  the  length  of 
its  shadow  diminished  by  20  feet.  Eequired  the  length  of  its 
shadow. 

83.  What  time  is  it  if  |  of  the  time  past  noon  is  equal  ta 
■^  of  the  time  to  midnight  ? 

84.  Between  2  and  3  o'clock  the  minute  and  hour  hands  of 
a  clock  are  together.     What  time  is  it  ? 


ARITHMETICAL   PROBLEMS  9 

85.  Which  weighs  the  more,  a  pound  of  feathers  or  a  pound 
of  gold  ? 

86.  Four  pedestrians  whose,  rates  are  as  the  numbers  2,  4, 
6,  and  8,  start  from  the  same  point  to  walk  in  the  same  direc- 
tion ai'ound  a  circular  tract  100  yards  in  circumference.  How 
far  has  each  gone  when  they  are  next  together  ? 

87.  If  2  miles  of  fence  will  inclose  a  square  of  160  acres, 
how  large  a  square  will  3  miles  of  fence  inclose  ? 

88.  I  bought  a  hojse  for  $  90,  sold  it  fon  $  100,  and  soon 
repurchased  it  for  $  80.     How  much  did  I  make  by  trading  ? 

89.  Considering  the  earth  8000,  and  the  sun  800,000  miles 
in  diameter,  how  many  earths  would  it  take  to  equal  the  sun  ? 

90.  A  merchant  marks  his  goods  to  sell  at  an  advance  of 
25%,  and  sells  a  book  for  $2.25,  and  allows  the  customer 
10  %  oif  from  the  marked  price.     What  did  the  book  cost  the 

merchant "/ 

91.  A  merchant  gives  a  discount  of  10%,  but  uses  a  yard 
measure  .72  of  an  inch  too  short.  What  rate  of  discount  would 
allow  him  the  same  amount  of  gain  if  the  measure  were  cor- 
rect? 

92.*  A  merchant  at  one  straight  cut  took  off  a  segment  of  a 
cheese  which  weighed  2  pounds,  and  had  \  of  the  circumfer- 
ence.    W^hat  was  the  weight  of  the  whole  cheese  ? 

93.  What  is  the  shortest  distance  that  a  fiy  will  have  to  go, 
crawling  from  one  of  the  lower  corners  of  the  room  to  the  op- 
posite upper  corner  —  the  room  being  20  feet  long,  15  feet 
wide,  and  10  high  ? 

94.  I  buy  goods  at  50  %  off  and  sell  them  at  40  and  10  % 
off.     What  is  my  per  cent  profit  ? 

95.  A  farmer  goes  to  a  store  and  says :  "  Give  me  as  much 
money  as  I  have  and  I  will  spend  ten  dollars  with  you."  It  is 
given  him,  and  the  farmer  repeats  the  operation  to  a  second, 


10  MATHEMATICAL  WRINKLES 

and  a  third  store,  and  has  no  money  left.     What  did  he  have 
in  the  beginning  ? 

96.  A  book  and  a  pen  cost  $1.20;  the  book  cost  $1  more 
than  the  pen.     What  was  the  cost  of  each  ? 

97.  A  dealer  asked  30%  profit,  but  sold  for  10  %  less  than 
he  asked.     What  per  cent  did  he  gain  ? 

98.  Suppose  we  leave  the  Pacific  coast  at  sunrise,  on  Sep- 
tember 28,  and  cross  the  Pacific  Ocean  fast  enough  to  have  sun- 
rise all  the  way  over  to  Manila,  where  it  is  sunrise  September 
29.     How  do  you  account  for  the  lost  day  ? 

99.  A  man  was  asked  whether  he  had  a  score  of  sheep.  He 
replied,  "  No,  but  if  I  had  as  many  more,  half  as  many  more, 
and  two  sheep  and  a  half,  I  should  have  a  score."  How  many 
had  he  ? 

100.  What  part  of  threepence  is  a  third  of  twopence  ? 

101.  Three  boys  met  a  servant  maid  carrying  apples  to 
market.  The  first  took  half  of  what  she  had,  but  returned  to 
her  10 ;  the  second  took  J,  but  returned  2 ;  and  the  third  took 
away  half  those  she  had  left,  but  returned  1.  She  then  had 
12  apples.     How  many  had  she  at  first  ? 

102.  A  person  having  about  him  a  certain  number  of 
German  coins,  said,  "If  the  third,  fourth,  and  sixth  of  them 
were  added  together,  they  would  make  54."  How  many  did 
he  have  ? 

103.  If  a  log  starts  from  the  source  of  a  river  on  Friday,  and 
floats  80  miles  down  the  stream  during  the  day,  but  comes 
back  40  miles  during  the  night  with  the  return  tide,  on 
what  day  of  the  week  will  it  reach  the  mouth  of  the  river, 
which  is  300  miles  long  ? 

104.  1x2x3x4x5x6x7x8x9x0=? 

105.  One  gentleman  meeting  another  and  inquiring  the  time 
past  12  o'clock,  received  for  an  answer,  "  One  third  of  the  time 
from  now  to  midnight."     What  time  in  the  afternoon  was  it  ? 


ARITHMETICAL  PROBLEMS  11 

106.  A  said  to  B,  "  Give  me  $  100,  and  then  I  shall  have  as 
much  as  you."  B  said  to  A,  "  Give  me  $  100,  and  then  I  shall 
have  twice  as  much  as  you."     How  many  dollars  had  each  ? 

107.  At  the  rate  of  4  miles  per  hour,  a  raft  floats  past  the 
lantliiig  at  8  A.M.;  the  down-going  steamer,  at  the  rate  of  16 
miles  per  hour,  passes  the  landing  at  4  p.m.  What  time  is  it 
when  the  steamer  overtakes  the  raft  ? 

108.  A  bought  a  horse  for  $  80  and  sold  it  to  B  at  a  certain 
rate  per  cent  of  gain.  B  sold  it  to  C  at  the  same  rate  per  cent 
of  gain.  C  paid  $105.80  for  the  horse.  What  price  did  B 
pay,  and  what  was  the  rate  per  cent  of  gain  ? 

109.  The  sum  of  two  numbers  is  582  and  their  difference  is 
218.     What  are  the  numbers  ? 

110.  What  are  the  contents  and  inside  surface  of  a  cubical 
box  whose  longest  inside  measurement  is  2  feet  ? 

111.  Three  persons  engaged  in  a  trade  with  a  joint  capital 
of  S  9000.  A's  capital  was  in  trade  5  months,  B's  2  months, 
and  C's  1  month  A's  share  of  the  gain  was  S  450,  B's  $  270, 
and  C's  $  180.     What  was  the  capital  of  each  ? 

112.  A  man  was  hired  for  a  year  for  $  100  and  a  suit  of 
clothes,  but  at  the  end  of  8  months  he  left  and  received  his 
clothes  and  $  60  in  money.  What  was  the  value  of  the  suit 
of  clothes  ? 

1 13.  A  note  for  $  100  was  due  on  September  1,  but  on  August 
11,  the  maker  proposed  to  pay  as  much  in  advance  as  would 
allow  him  60  days  after  September  1,  to  pay  the  balance. 
How  much  did  he  pay  August  11,  money  being  worth  6  %  ? 

114.  If  I  rent  a  house  at  $18  a  month,  payable  monthly  in 
advance,  what  amount  of  cash  payable  at  the  beginning  of  the 
year  will  pay  the  year's  rent,  interest  at  5  %  ? 

115.  If  a  house  rents  for  $20  a  month,  payable  at  the  close 
of  each  month,  what  amount  is  due  if  not  paid  till  the  end  of 
year,  interest  at  6  %  ? 


12  MATHEMATICAL   WEINKLES 

116.  A  merchant  sold  a  lease  of  $480  a  year,  payable  quar- 
terly, having  8  years  and  9  months  to  rim,  for  $  2500.  Did  he 
gain  or  lose,  and  how  much,  interest  at  8%,  payable  semi- 
annually ? 

117.  A  box  of  oranges  weighed  64  pounds  by  the  grocer's 
scales,  but  being  placed  in  the  other  scale  of  the  balance,  it 
weighed  only  30  pounds.  What  was  the  true  weight  of  the 
box  of  oranges  ? 

118.  If  a  ball  5  inches  in  diameter  weighs  8  pounds,  what 
will  be  the  weight  of  a  similar  ball  10  inches  in  diameter  ? 

119.  A,  B,  and  C  dine  on  8  loaves  of  bread.  A  furnishes  5 
loaves,  B  3  loaves,  and  C  pays  the  others  8  cents  for  his  share. 
How  must  A  and  B  divide  the  money  ? 

120.  A  boy  being  asked  how  many  fish  he  had,  replied,  "  11 
fish  are  7  more  than  -|  of  the  number."     How  many  had  he  ? 

121.  I  have  two  lamps,  one  of  4-candle  power,  and  one  of 
9-candle  power.  If  the  former  is  30  feet  distant,  how  far 
away  must  I  place  the  latter  to  give  me  the  same  amount 
of  light? 

122.  A  merchant  bought  90  boxes  of  lemons  for  $265,  pay- 
ing $  3.50  for  first  quality  and  $  3  for  second  quality.  How 
many  boxes  of  each  kind  did  he  buy  ? 

123.  A  vessel  after  sailing  due  north  and  due  east  on  alter- 
nate days,  is  found  to  be  16V2  miles  northeast  of  the  starting 
place.     What  distance  has  it  sailed  ? 

124.  Two  teachers  work  together ;  for  10  days'  work  of  the 
first  and  8  days'  work  of  the  second  they  receive  $28,  and  for 
5  days'  work  of  the  first  and  11  days'  work  of  the  second  they 
receive  $21.     What  is  each  man's  daily  wages? 

125.  A  hind  wheel  of  a  carriage  4  feet  6  inches  high  re- 
volved 720  times  in  going  a  certain  distance.  How  many 
revolutions  did  the  fore  wheel  make,  which  was  4  feet  high  ? 


ARITHMETICAL  PROBLEMS  13 

126.  A  farmer  carried  some  eggs  to  market,  for  which  he 
received  $  2.56,  receiving  as  many  cents  a  dozen  as  there  were 
dozen.     How  many  dozen  were  there  ? 

127.  Three  men,  A,  B,  and  C,  ai-e  to  mow  a  circular  meadow 
containing  9  acres.  A  is  to  receive  $3,  B  S4,  and  C$5  for 
his  work.     What  width  must  each  man  mow  ? 

128.  If  the  diameter  of  a  cannon  ball  is  100  times  that  of 
a  bullet,  how  many  bullets  will  it  take  to  equal  the  cannon 
ball? 

129.  A  man  sells  a  cow  and  a  horse  for  $  120.  He  sells  the 
horse  for  $100  more  than  the  cow.     What  did  he  sell  each  for? 

130.  If  a  man  5J  feet  tall  weighs  166.375  pounds,  how 
much  will  a  man  6  feet  tall  of   similar   proportions  weigh  ? 

131.  Having  sold  a  house  and  lot  at  4  %  commission,  I  in- 
vest the  net  proceeds  in  merchandise  after  deducting  my  com- 
mission of  2%  for  buying.  My  whole  commission  is  $50. 
For  how  much  did  I  sell  the  house  and  lot  ? 

132.  A  teacher  agreed  to  teach  a  10-weeks  school  for  $  100 
and  his  board.  At  the  end  of  the  term,  on  account  of  3 
weeks'  absence  caused  by  sickness,  he  received  only  $58. 
What  was  his  board  per  week  ? 

133.  In  buying  a  bill  of  goods,  I  am  offered  my  choice  of 
50,  25,  and  5  %  discount,  or  5,  25,  and  50  %  discount.  Which 
is  better? 

134.  The  product  of  two  numbers  exceeds  their  difference 
by  their  sum.     Find  one  of  the  numbers. 

135.  Twice  the  sum  of  two  numbers  plus  twice  their  differ- 
ence is  80.     What  is  the  greater  number  ? 

136.  One  half  the  sum  of  two  numbers  exceeds  one  half 
their  difference  by  60.     What  is  the  smaller  number? 

137.  What  per  cent  is  gained  by  sellir^g  13  ounces  of  coffee 
for  a  pound  ? 


14  MATHEMATICAL  WKIKKLES 

138.  If  I  sell  I  of  an  acre  of  land  for  what  an  acre  cost  me, 
what  per  cent  do  I  gain  ? 

139.  I  sold  a  horse  for  $  200,  losing  20  %  ;  I  bought  another 
and  sold  it  at  a  gain  of  25  %  ;  I  neither  gained  nor  lost  on  the 
two.    What  was  the  cost  of  each  ? 

140.  At  the  time  of  marriage  a  wife's  age  was  f  of  the  age  of 
her  husband,  and  24  years  after  marriage  her  age  was  \^  of  the 
age  of  her  husband.    How  old  was  each  at  the  time  of  marriage  ? 

141.  How  much  water  is  there  in  a  mixture  of  50  gallons  of 
wine  and  water,  worth  $  2  per  gallon,  if  50  gallons  of  the  wine 
costs  $250? 

142.  A  Texas  farmer  keeps  2100  cows  on  his  farm.  For 
every  3  cows  he  plows  1  acre  of  ground  and  for  every  7  cows 
he  pastures 2  acres  of  land.     How  many  acres  are  in  his  farm? 

143.  The  divisor  is  6  times  the  quotient.    Find  the  quotient. 

144.  When  gold  was  worth  25  %  more  than  paper  money, 
what  was  the  value  in  gold  of  a  dollar  bill  ? 

145.  I  bought  15  yards  of  ribbon,  and  sold  10  of  them  for 
what  I  paid  for  all,  and  the  remainder  at  cost.  I  gained  $  .25 
by  the  transaction.     What  did  the  ribbon  cost  me  ? 

146.  If  a  ball  of  yarn  6  inches  in  diameter  makes  one  pair 
of  gloves,  how  many  similar  pairs  will  a  ball  12  inches  in 
diameter  make  ? 

147.  At  what  time  between  4  and  5  o'clock  do  the  hour  and 
minute  hands  of  a  clock  coincide  ? 

148.  At  what  time  between  2  and  3  o'clock  do  the  hour  and 
minute  hands  of  a  clock  coincide  ? 

149.  At  what  time  between  2  and  3  o'clock  are  the  hour  and 
minute  hands  of  the  clock  at  right  angles  ? 

150.  At  what  time  between  2  and  3  o'clock  are  the  hands 
of  a  clock  exactly  opposite  each  other  ? 


ARITHMETICAL  PROBLEMS  15 

151.  From  200  hundredths  take  15  tenths. 

152.  Find  the  sum  of  2324  thousandths  and  24,325  hun- 
dredths. 

153.  A  lady  at  her  marriage  had  her  husband  agree  that  if 
at  his  death  they  should  have  only  a  daughter,  she  should  have 
J  of  his  estate ;  and  if  they  should  have  only  a  son,  she  should 
have  |.  They  had  a  son  and  a  daughter.  How  much  should 
each  receive,  if  the  estate  was  worth  $  23,375  ? 

154.  A  crew  can  row  24  miles  downstream  in  3  hours,  but 
requires  4  hours  to  row  back.    What  is  the  rate  of  the  current? 

155.  What  minuend  is  80  greater  than  the  subtrahend,  which 
is  20  greater  than  the  remainder  ? 

156.  The  G.  C.  D.  of  two  numbers  is  60  and  the  L.  C.  M.  is 
720.     Find  the  product  of  the  numbers. 

157.  In  extracting  the  cube  root  of  a  perfect  power  the  oper- 
ator found  the  last  complete  dividend  to  be  132,867.  Find  the 
power. 

158.  A  merchant  marks  his  goods  at  an  advance  of  25  %  on 
cost.  After  selling  J  of  the  goods,  he  finds  that  some  of  the 
goods  on  hand  are  damaged  so  as  to  be  worthless ;  he  marks 
the  salable  goods  at  an  advance  of  10  %  on  the  marked  price 
and  finds  in  the  end  that  he  has  made  20  %  on  cost.  What 
part  of  the  goods  was  damaged  ? 

159.  A  king  has  a  horse  shod  and  agrees  to  pay  1  cent  for 
driving  the  first  nail,  2  cents  for  the  second,  4  cents  for  the 
third,  doubling  each  time.  What  will  the  shoeing  with  32 
nails  cost? 

160.  I  sold  a  book  at  a  loss  of  25  %.  Had  it  cost  me  $1 
more,  my  loss  would  have  been  40%.     Find  its  cost. 

161.  At  noon  the  three  hands  —  hour,  minute,  and  second  — 
of  a  clock  are  together.  At  what  time  will  they  first  be  to- 
gether again? 


16  MATHEMATICAL   WKINKi.ES 

162.  A  train  is  traveling  from  one  station  to  another.  After 
traveling  an  hour  it  breaks  down  and  is  delayed  for  an  hour. 
It  then  proceeds  at  f  of  its  former  speed,  and  arrives  3  hours 
late.  Had  it  gone  50  miles  farther  before  the  breakdown,  it 
would  have  arrived  1  hour  and  20  minutes  sooner.  Find  the 
rate  of  the  train  and  the  distance  between  the  stations. 

163.  If  a  cocoanut  4  inches  in  diameter  is  worth  5  cents, 
what  is  the  worth  of  one  6  inches  in  diameter  ? 

164.  Prove  that  the  product  of  the  G.C.D.  and  L.C.M.  of 
two  numbers  is  equal  to  the  product  of  the  numbers. 

165.  Sum  to  infinity  the  series  l  +  Y+i  +  B'+  **•• 

166.  Find  the  sum  of  1  +  i  +  i  +  2V  +  •  •  •  to  infinity. 

167.  Find  the  sum  of  4  -f-  0.4  -{-  0.04  +  ...  to  infinity. 

168.  What  is  the  distance  passed  through  by  a  ball  before 
it  comes  to  rest,  if  it  falls  from  a  height  of  100  feet  and  re- 
bounds half  the  distance  at  each  fall  ? 

169.  Two  trains  start  at  the  same  time,  ouq  from  Jackson- 
ville to  Savannah,  the  other  from  Savannah  to  Jacksonville. 
If  they  arrive  at  destinations  1  hour  and  4  hours  after  passing, 
what  are  their  relative  rates  of  running  ? 

170.  If  sound  travels  at  the  rate  of  1090  feet  per  second, 
how  far  distant  is  a  thundercloud  when  the  sound  of  the  thun- 
der follows  the  flash  of  lightning  after  10  seconds  ? 

171.  The  G.C.D.  and  the  L.C.M.  of  two  numbers  between 
100  and  200  are  respectively  4  and  4620.     Find  the  numbers. 

172.  What  three  equal  successive  discounts  are  equivalent 
to  a  single  discount  of  58.8  %  ? 

173.  How  much  will  the  product  of  two  numbers  be  in- 
creased by  increasing  each  of  the  numbers  by  1  ? 

174.  I  can  beat  James  4  yards  in  a  race  of  100  yards,  and 
James  can  beat  John  10  yards  in  a  race  of  200  yards.  How 
many  yards  can  I  beat  John  in  a  race  of  500  yards  ? 


a:^thmetical  problems  17 

175.  Three  ladies  own  a  ball  of  yarn  G  inches  in  diameter. 
What  portion  of  the  diameter  must  each  wind  off  in  order  to 
divide  the  yarn  equally  among  them  ? 

176.  Demonstrate  the  following :  If  the  greater  of  two  num- 
bers is  divided  by  the  less,  and  the  less  is  divided  by  the 
remainder,  and  this  process  is  continued  till  there  is  no  re- 
mainder, the  last  divisor  will  be  the  greatest  common  divisor. 

177.  Find  the  volume  of  a  rectangular  piece  of  ice  8  feet 
long,  7  feet  wide,  and  floating  in  water,  with  2.4  inches  of  its 
thickness  above  water,  the  specific  gravity  of  ice  being  .9. 

178.  Two  trains,  400  and  200  feet  long  respectively,  are 
moving  with  uniform  velocities  on  parallel  rails;  when  they 
move  in  opposite  directions  they  pass  each  other  in  5  seconds, 
but  when  they  move  in  the  same  direction,  the  faster  train 
passes  the  other  in  15  seconds.  Find  the  rate  per  hour  at 
which  each  train  moves. 

179.  A  boy  is  running  on  a  horizontal  plane  directly  towards 
the  foot  of  a  tree  50  feet  in  height.  When  he  is  100  feet  from 
the  foot  of  the  tree,  how  much  faster  is  he  approaching  it  than 
the  top  ? 

180.  Express  77,610  in  the  duodecimal  scale. 
181.*   In  what  scale  is  6  times  7  expressed  by  110? 

182.  Express  Adam's  age  at  his  death  in  the  binary  scale. 

183.  Add  3152e,  4204e,  3241e,  SlOSg. 

184.  Subtract  12,3125  from  23,024^. 

185.  Multiply  62,453;  by  325;. 

186.  Divide  2,034,431,  by  234,. 

187.  Extract  the  square  root  of  170». 

188*    Extract  the  cube  root  of  3I2O4. 

189.  How  many  trees  can  be  set  out  upon  a  space  100  feet 
square,  allowing  no  two  to  be  nearer  each  other  than  10  feet  ? 


18  MATHEMATICAL  WKINKLES 

190.  How  many  stakes  can  be  driven  down  upon  a  space  12 
feet  square,  allowing  no  two  to  be  nearer  each  other  than  1 
foot? 

191.  Multiply  789,627  by  834,  beginning  at  the  left  to 
multiply. 

192.  Two  fifths  of  a  mixture  of  wine  and  water  is  wine ;  but 
if  10  gallons  of  water  be  added  to  it,  then  only  -^-^  of  the  mix- 
ture will  be  wine.  How  many  gallons  of  each  liquid  is  in  the 
mixture  ? 

193.  Simplify 
10  4- — 


1  +  -1 


1  + 


1-i 


194.  15,600  is  the  product  of  three  consecutive  numbers. 
What  are  they  ? 

195.  Find  a  number  which  is  as  much  greater  than  1042  as 
it  is  less  than  1236. 

196.  Multiply  729,038  by  105,357  using  only  3  multipliers. 

197.  What  is  the  smallest  number  to  be  subtracted  from 
10,697  to  make  the  result  a  perfect  cube  ? 

198.  I  wish  to  reach  a  certain  place  at  a  certain  time ;  if  I 
walk  at  the  rate  of  4  miles  an  hour,  I  shall  be  10  minutes  late, 
but  if  I  walk  5  miles  an  hour,  I  shall  be  20  minutes  too  soon. 
How  far  have  I  to  walk  ? 

199.  A  wineglass  is  half  full  of  wine,  and  another  twice 
the  size  is  \  full.  They  are  then  filled  up  with  water,  and  the 
contents  mixed.  What  part  of  the  mixture  is  wine,  and  what 
part  water  ? 

200.  A  cork  globe  2  feet  in  diameter,  whose  specific  gravity 
is  -Jg,  is  hollowed  out  and  filled  with  lead  whose  specific 
gravity  is  10.  What  must  be  the  thickness  of  the  shell  of  cork 
so  that  it  will  sink  just  even  with  the  surface  of  the  water? 


ARITHMETICAL  PROBLEMS  19 

201.  What  temperature  will  result  from  mixing  100  pounds 
of  ice  at  14°  F.  with  80  pounds  of  steam  at  270°  F.  ? 

202.  It  is  1800  miles  from  A  to  C,  and  the  "  Sunset  Flyer  " 
annihilates  the  distance  in  50  hours.  She  averages  30  miles 
an  hour  from  A  to  B,  and  55  miles  an  hour  from  B  to  C. 
Locate  B. 

203.  A  square  and  its  circumscribing  circle  revolve  about 
the  diagonal  of  the  square  as  an  axis.  Compare  the  volumes 
and  surfaces  of  the  solids  generated,  the  diagonal  being  6  feet. 

204.  The  aggregate  area  of  two  square  fields  is  8J  acres. 
The  side  of  the  second  is  10  rods  longer  than  that  of  the  first. 
Ascertain  the  length  of  the  first. 

205.  How  high  above  the  earth's  surface  (radius  4000  miles) 
would  a  pound  weight  weigh  but  one  ounce  avoirdupois  by 
a  scale  indicator,  corrected  for  change  of  elasticity  by  tem- 
perature ? 

206.  On  a  west-bound  freight  train  a  man  is  running  east- 
ward at  the  rate  of  6  miles  an  hour,  and  likewise  a  man  runs 
in  the  same  direction  8  miles  an  hour  on  a  train  going  east. 
If  the  trains  pass  while  running  36  and  22  miles  an  hour,  re- 
spectively, how  many  miles  apart  are  the  men  at  the  end  of 
one  minute  from  the  moment  they  pass  each  other  ? 

207.  A  drawer  made  of  inch  boards  is  8  inches  wide,  6 
inches  deep,  and  slides  horizontally.  How  far  must  it  be 
drawn  out  to  put  into  it  a  book  4  inches  wide  and  9  inches 
long? 

208.  The  dividend  is  4352,  the  remainder  17,  which  is  the 
G.C.  D.  of  the  quotient  and  divisor,  whose  difference  you  may 
find. 

209.  B  paid  S9  more  than  true  discount  by  borrowing 
money  at  a  bank  for  one  year  at  12  % .  Find  the  face  of  the 
note. 


20  MATHEMATICAL  WRINKLES 

210.  How  many  feet  of  inch  lumber  in  a  wagon  tongue  10 
feet  long,  4  inches  square  at  one  end  and  2  inches  by  3  inches 
at  the  other  end  ? 

211.  How  many  inch  balls  can  be  put  in  a  box  which  meas- 
ures inside  10  inches  square  and  5  inches  deep  ? 

212.  If  the  posts  of  a  wire  fence  around  a  rectangular  field 
twice  as  long  as  wide  were  set  16  feet  apart  instead  of  12  feet, 
it  would  save  66  posts.     How  many  acres  in  the  field  ? 

213.  If  gold  is  19.3  times  as  heavy  as  water  and  copper  8.89 
as  heavy,  how  many  times  as  heavy  is  a  coin  composed  of  11 
parts  of  gold  and  1  part  of  copper  ? 

214.  A  ball  falls  15  feet  and  bounces  back  5  feet.  How  far 
will  it  bound  before  it  comes  to  rest  ? 

215.  A  borrows  $500  from  a  building  and  loan  association 
and  agrees  to  pay  $9.50  per  month  for  72  months,  the  first 
payment  to  be  made  at  the  end  of  the  first  month.  What  rate 
of  interest  does  he  pay  ?  The  association  claims  to  charge 
only  8  %  (the  legal  rate  in  Alabama).  How  can  the  per  cent 
be  figured  out  ? 

216.  A  rope  50  feet  long  is  fastened  to  two  stakes,  driven  40 
feet  apart.  A  calf  is  fastened  to  a  ring  which  moves  freely  on 
this  rope.     Over  what  area  can  the  calf  graze  ? 

217.  A  metal  dog  made  of  gold  and  silver  weighs  8.75 
ounces.  Its  specific  gravity  is  14.625,  that  of  gold  19.25,  and 
that  of  silver  10.5.     Find  the  number  of  ounces  of  gold  in  it. 

218.  By  drilling  an  inch  hole  through  a  cubical  block  of 
wood  parallel  to  the  faces  of  the  block,  -J^  of  the  wood  was 
cut  away.    What  were  the  dimensions  of  the  block  ? 

219.  Find  two  numbers  whose  G.  CD.  is  24,  and  L.  C.  M. 

288. 

220.  Find  the  greatest  number  that  will  divide  364,  414, 
and  539,  and  leave  the  same  remainder  in  each  case. 


ARITHMETICAL  PROBLEMS  21 

221.  Had  an  article  cost  me  8%  less,  the  number  of  per 
cent  gain  would  have  been  10  %  more.     What  was  the  gain  ? 

222.  At  what  time  between  3  and  4  o'clock  will  the  minute 
hand  be  as  far  from  12  on  the  left  side  of  the  dial  plate  as 
the  hour  hand  is  from  12  on  the  right  side  ? 

223.  A  ball  whose  specific  gravity  is  3|  measures  a  foot  in 
diameter.  Find  the  diameter  of  another  ball  of  the  same 
weight  but  with  a  specific  gravity  of  2J^. 

224.  A  owes  $  2500  due  in  two  years.  He  pays  $  500  cash 
and  gives  a  note  payable  in  8  months,  for  the  balance.  Find 
the  face  of  the  note,  money  being  worth  6  %. 

225.  A  man  bought  a  horse  for  $201,  giving  his  note  due 
in  30  days.  He  at  once  sold  the  horse,  taking  a  note  for 
$224.40,  due  in  4  months.  What  was  his  rate  of  gain  at  the 
time  of  the  sale,  interest  6  %  ? 

226.  The  minute  hand  and  the  hour  hand  coincide  every  65 
minutes.     Does  the  clock  gain  or  lose,  and  how  much  ? 

227.  A  ball  weighing  970  ounces,  weighs  in  water  892 
ounces,  and  in  alcohol  910  ounces.  What  is  the  specific 
gravity  of  alcohol  ? 

228.  A  steamer  moves  through  8°  of  longitude  daily  in  ply- 
ing to  and  fro  across  the  Atlantic.  How  long  is  it  from  one 
noon  to  the  next  ? 

229.  A,  B  and  C  raise  165  acres  of  grain.  A  owns  100  acres 
of  the  land  and  B  65  acres.  C  pays  the  others  $110  rent. 
How  must  A  and  B  divide  this  money  if  the  grain  is  shared 
equally  ? 

230.  A  silver  cup  is  a  hemisphere  filled  with  wine  worth 
$1.20  a  quart.  The  value  of  the  cup  is  2  dimes  for  every 
square  inch  of  internal  surface,  and  the  cup  is  worth  just  as 
much  as  the  wine.     What  is  the  value  of  the  cup  ? 

231.  A  ball  12  inches  in  diameter  is  rolled  around  a  circular 
room  12  feet  in  diameter  in  such  a  way  that  it  always  touches 


22  MATHEMATICAL   WRINKLES 

both  wall  and  floor.     How   many  revolutions    does   the   ball 
make  in  rolling  once  around  the  room  ? 

232.  A  man  desires  to  purchase  eggs  at  5  cents,  1  cent, 
and  ^  cent,  respectively,  in  such  numbers  that  he  will  obtain 
100  eggs  for  a  dollar.  How  many  solutions  in  rational  inte- 
gers ? 

233.  How  many  board  feet  in  a  piece  of  lumber,  2  inches 
square  at  one  end  and  at  the  other  end  1  inch  by  12  inches, 
if  the  ends  are  parallel  ? 

234.  How  many  board  feet  in  the  above  piece  of  lumber  if 
it  is  24  feet  long  ? 

235.  Is  anything  expressed  by  .^  ?     If  so,  what  ? 

236.  A  man  bequeathed  to  his  son  all  the  land  he  could  in- 
close in  the  form  of  a  right-angled  triangle  with  2  miles  of 
fence,  the  base  of  the  triangle  to  be  128  rods.  How  many 
acres  did  he  get? 

237.  The  distance  around  a  rectangular  field  is  140  rods, 
and  the  diagonal  is  50  rods.  Find  its  length,  breadth,  and 
area. 

238.  The  specific  gravity  of  ice  being  .918  and  of  sea  water 
1.03,  find  the  volume  of  an  iceberg  floating  with  700  cubic 
yards  above  water. 

239.  A  room  is  30  feet  long,  12  feet  wide,  and  12  feet  high. 
At  one  end  of  the  room,  3  feet  from  the  floor,  and  midway 
from  the  sides,  is  a  spider.  At  the  other  end,  9  feet  from  the 
floor,  and  midway  from  the  sides,  is  a  fly.  Determine  the 
shortest  path  by  way  of  the  floor,  ends,  sides,  and  ceiling, 
the  spider  can  take  to  capture  the  fly. 

240.  A  and  B  are  engaged  in  buying  hogs,  each  paying  out 
of  his  individual  funds  for  hogs  purchased  by  him,  and  each 
retaining  as  his  individual  funds  the  money  received  from  sales 
made  by  him.     They  now  wish  to  form  a  partnership  to  cover 


ARITHMETICAL  PROBLEMS  23 

all  past  transactions  and  to  share  equally  in  the  settlement  for 
sales  and  purchases,  and  also  to  be  equally  interested  in  hogs 
which  they  have  on  hand  unsold.     The  following  data  given : 

A  has  paid  for  hogs  $1183.35,  and  received  from  sales  of 
hogs  $434.35.  ' 

B  has  paid  for  hogs  $241.55,  and  received  from  sales  of  hogs 
$619.00. 

Invoice  of  hogs  on  hand  at  this  time  $511.35. 

How  much  does  A  owe  B,  or  B  owe  A,  so  that  they  will  have 
shared  equally  in  payments  and  receipts,  and  be  equally  inter- 
ested in  the  hogs  on  hand  ? 

241.  The  hour,  minute,  and  second  hands  of  a  clock  turn  on 
the  same  center.  At  what  time  after  12  o'clock  is  the  hour 
hand  midway  between  the  other  two  ?  The  second  hand  mid- 
way between  the  other  two?  The  minute  hand  midway  be- 
tween the  other  two  ? 

242.  My  agent  sold  pork  at  a  commission  of  7%.  The  pro- 
ceeds being  increased  by  $6.20,  I  ordered  him  to  buy  cattle 
at  a  commission  of  3J%.  Cattle  now  declined  in  price  33 J  %, 
and  I  found  my  total  loss,  including  commissions,  to  be  exactly 
$1002.20.     Find  the  value  of  the  pork. 

243.  A  owes  $900,  due  December  10,  but  he  makes  two  equi- 
table payments,  one  September  8  and  the  other  January  10. 
Find  each  payment. 

244.  A  man,  dying,  left  an  estate  of  $23,480  to  his  three 
sons,  aged  15,  13,  and  11  years,  to  be  so  divided  that  each  share 
placed  at  interest  shall  amount  to  the  same  sum  as  the  sons, 
respectively,  become  21  years  of  age.  What  was  each  son's 
share,  money  being  worth  5  %  ? 

245.  A  man  spent  $  100  in  buying  two  kinds  of  silk  at  $  4.50 
and  $4.00  a  yard;  by  selling  it  at  $4.25  per  yard  he  gained 
2  ojo  •     How  many  yards  of  each  did  he  buy  ? 


24  MATHEMATICAL   WKIKKLES 

246.  A  lady  being  asked  the  time  of  day  replied,  "It  is 
between  4  and  5  o'clock,  and  the  hour  and  minute  hands  are 
together."     What  was  the  time  ? 

247.  Three  men  A,  B,  and  C  can  do  a  piece  of  work  in  60 
days.  After  working  together  10  days,  A  withdraws  and  B 
and  C  work  together  at  the  same  rate  for  20  days,  then  B  with- 
draws and  C  completes  the  work  in  96  days,  working  i  longer 
each  day.  Working  at  his  former  rate,  C  alone  could  do  the 
work  in  222  days.  Find  how  long  it  would  take  A  and  B  each 
separately  to  do  the  work. 

248.  In  a  class  there  are  twice  as  many  girls  as  boys.  Each 
girl  makes  a  bow  to  every  other  girl,  to  every  boy,  and  to  the 
teacher.  Each  boy  makes  a  bow  to  every  other  boy,  to  every 
girl,  and  to  the  teacher.  In  all  there  are  900  bows  made. 
How  many  boys  are  in  the  class  ? 

249.  A  boy  weighing  96  pounds  is  seated  on  one  end  of  a  see- 
saw 16  feet  long,  and  a  boy  weighing  120  pounds  is  seated  on 
the  other  end.  Find  the  distance  of  each  boy  from  the  point 
of  support,  the  lengths  of  the  two  arms  of  the  plank  being 
inversely  proportional  to  the  weights  at  their  ends. 

250.  Two  men  are  on  opposite  sides  of  the  center  of  the 
earth.  Find  the  shortest  distance  that  each  will  be  required 
to  go  in  order  to  exchange  places,  provided  they  travel  different 
routes  and  so  travel  as  to  enjoy  each  other's  company  for  500 
miles  of  the  distance.     (Radius  of  earth  =  4000  miles.) 

251.  A  conical  wine  glass  2  inches  in  diameter  and  3 
inches  deep  is  ^  full  of  water.  What  is  the  depth  of  the 
water  ? 

252.  A  hollow  sphere  8  inches  in  diameter  is  filled  with 
water.  How  many  hollow  cones,  each  8  inches  in  altitude, 
and  8  inches  in  diameter  at  the  base,  can  be  filled  with  the 
water  in  the  sphere  ? 


ALGEBRAIC   PROBLEMS 

1.  I  am  now  twice  as  old  as  you  were  when  I  was  your 
age.  When  you  are  as  old  as  I  now  am,  the  sum  of  our  ages 
will  be  100.     What  are  our  ages  ? 

2.  A  starts  from  Gunter  to  Denton,  and  at  the  same  time 
B  starts  from  Denton  to  Gunter ;  A  reaches  Denton  32  hours, 
and  B  reaches  Gunter  60  hours,  after  they  meet  on  the  way. 
In  how  many  hours  do  they  make  the  journey? 

3.  At  what  time  between  10  and  11  o'clock  is  the  second 
hand  of  a  clock  one  minute  space  nearer  to  the  hour  hand  than 
it  is  to  the  minute  hand? 

4.  In  walking  along  a  street  on  which  electric  cars  are 
running  at  equal  intervals  from  both  ends,  I  observe  that  I 
am  overtaken  by  a  car  every  12  minutes,  and  that  I  meet  one 
every  4  minutes.  What  are  the  relative  rates  of  myself  and 
tjie  cars,  and  at  what  intervals  of  time  do  the  cars  start  ? 

6.  What  are  eggs  per  dozen  when  2  less  in  a  shilling's 
worth  raise  the  price  one  penny  per  dozen? 

6.  Two  men  agree  to  build  a  walk  100  yards  in  length  for 
S200.  They  divide  the  work  so  that  one  man  should  receive 
60  cents  more  per  yard  than  the  other.  How  many  yards 
does  each  man  build,  if  he  receives  $100? 

7.  Two  boats  start  from  opposite  sides  of  a  river  at  the 
same  instant,  and  throughout  the  journeys  to  be  described 
maintain  their  respective  speed.  They  pass  one  another  at  a 
point  just  720  yards  from  the  left  shore.  Continuing  on  their 
respective  journeys,  they  reach  opposite  banks,  where  each 
boat  remains  10  minutes  and  then  proceeds  on  its  return  trip. 


26  MATHEMATICAL   WRINKLES 

This  time  the  boats  meet  at  a  point  400  yards  from  the  right 
shore.     What  is  the  width  of  the  river  ? 

8.  How  many  acres  does  a  square  tract  of  land  contain, 
which  sells  for  $  160  an  acre,  and  is  paid  for  by  the  number 
of  silver  dollars  that  will  lie  upon  its  boundary  ? 

9.  Two  girls,  4  feet  apart,  walk  side  by  side  around  a 
circular  park.  How  far  does  each  walk  if  the  sum  of  their 
distances  is  1  mile  ? 

10.  How  many  acres  are  there  in  a  field,  the  number  of 
rails  used  in  fencing  the  field  equaling  the  number  of  acres  — 
each  rail  being  11  feet  long  and  the  fence  4  rails  high  ? 

11.  Three  men  are  going  to  make  a  journey  of  40  miles. 
The  first  can  walk  at  the  rate  of  1  mile  per  hour,  the  second 
walks  at  the  rate  of  2  miles  per  hour,  and  the  third  goes  in  a 
buggy  at  the  rate  of  8  miles  per  hour.  The  third  takes  the 
first  with  him  and  carries  him  to  such  a  point  as  will  allow 
the  third  time  to  drive  back  to  meet  the  second,  and  carry  him 
the  remaining  part  of  the  40  miles,  so  as  all  may  arrive  at  the 
same  time.     How  long  will  it  require  to  make  the  journey  ? 

12.  Two  trains,  400  and  200  feet  long  respectively,  are  mov- 
ing with  uniform  velocities  on  parallel  rails ;  when  they  move 
in  opposite  directions,  they  pass  each  other  in  5  seconds,  but 
when  they  move  in  the  same  direction,  the  faster  train  passes 
the  other  in  15  seconds.  Find  the  rate  per  hour  at  which  each 
train  moves. 

13.  How  many  minutes  is  it  until  6  o'clock,  if  50  minutes 
ago  it  was  4  times  as  many  minutes  past  3  o'clock? 

14.  A  man  bought  a  gun  for  a  certain  price.  Now,  if  he 
sells  it  for  $  9,  he  will  lose  as  much  per  cent  as  the  gun  cost. 
Required  the  cost  of  the  gun. 

15.  In  a  nest  were  a  certain  number  of  eggs;  if  I  had 
brought  1  egg  that  I  didn't  bring,  I  should  have  brought  |  of 


ALGEBRAIC  PROBLEMS  27 

thera,  and  if  I  had  left  2  eggs  that  I  did  bring,  I  should  have 
brought  half  of  them.     How  many  eggs  were  in  the  nest? 

16.  A  man  sold  a  lot  for  $  144.  The  number  of  dollars  the 
lot  cost  was  the  same  as  the  number  of  per  cent  profit.  What 
did  the  lot  cost  ? 

17.  What  is  the  side  of  a  cube  which  contains  as  many  cubic 
inches  as  there  are  square  inches  in  its  surface  ? 

18.  What  is  the  length  of  one  edge  of  that  cube  which  con- 
tains as  many  solid  units  as  there  are  linear  units  in  the  diag- 
onal through  the  opposite  corners  ? 

19.  The  sum,  the  product,  and  the  difference  of  the  squares 
of  two  numbers  are  all  equal.     Find  the  numbers. 

20.  Upon  inquiring  the  time  of  day,  a  gentleman  was  in- 
formed that  the  hour  and  minute  hands  were  together  between 
4  and  5.     What  was  the  time  of  day  ? 

21.  An  officer  wishing  to  arrange  his  men  in  a  solid  square, 
found  by  his  first  arrangement  that  he  had  39  men  over.  He 
then  increased  the  number  of  men  on  a  side  by  1,  and  found 
50  men  were  needed  to  complete  the  square.  How  many  men 
did  he  have  ? 

22.  A  young  lady  being  asked  what  she  paid  for  her  eggs, 
replied,  "Three  dozen  cost  as  many  cents  as  I  can  buy  eggs  for 
36  cents."     What  was  the  price  per  dozen  ? 

23.  A  cube  is  formed  out  of  a  lot  of  cubical  blocks,  1  foot 
each,  and  it  is  found  by  using  448  more  another  cube  is  formed, 
the  edge  of  which  is  8  feet.  What  was  the  length  of  an  edge 
of  the  original  cube  ? 

24.  Find  two  numbers  whose  product  is  equal  to  the  differ- 
ence of  their  squares,  and  the  sum  of  their  squares  equal  to  the 
difference  of  their  cubes. 

25.  A  young  lady  being  asked  her  age,  answered,  "  If  you 
add  the  square  root  of  my  age  to  |  of  my  age,  the  sum  will  be 
10."    Required  her  age. 


28  MATHEMATICAL   WRINKLES 

26.  There  is  a  fish  whose  head  is  9  inches  long ;  the  tail  is 
as  long  as  the  head  and  |  the  body ;  and  the  body  is  as  long  as 
the  head  and  the  tail  together.  What  is  the  length  of  the 
fish? 

27.  I  bought  2  horses  for  S  80 ;  I  sold  them  for  $  80  apiece, 
the  gain  on  the  one  being  20  %  more  than  on  the  other.  What 
was  the  cost  of  each  ? 

28.  A  man  has  a  square  lot  upon  which  he  wishes  to 
build  a  house  facing  the  street,  with  a  driveway  around  the 
other  three  sides.  He  wants  the  house  to  cover  the  same 
amount  of  land  as  the  driveway.  How  wide  shall  he  make 
the  driveway,  the  lot  being  100  feet  each  way  ? 

29.  An  officer  can  form  his  men  into  a  hollow  square  4  deep, 
and  also  into  a  hollow  square  6  deep ;  the  front  in  the  latter 
formation  contains  12  men  fewer  than  in  the  former  formation. 
Find  the  number  of  men. 

30.  How  must  a  line  12  inches  long  be  divided  into  two 
parts  so  that  the  rectangle  of  the  whole  line  and  one  part  shall 
equal  the  square  on  the  other  side  ? 

31.  Two  miners,  B  and  C,  have  the  same  monthly  wages. 
B  is  employed  7  months  in  the  year,  and  his  annual  expenses 
are  $350;  C  is  employed  5  months  in  the  year,  and  his  annual 
expenses  are  $250.  In  5  years  B  saves  the  same  amount 
that  C  saves  in  7  years.  What  were  the  monthly  wages  of 
each? 


32. 

Simplify:    '^       ,  :            • 

33. 

Find  the  value  of  x  in  the  equation : 

2(l  +  ^')  =  (l  +  a;)^ 

34. 

Solve  the  equation : 

iB*  +  4  m^x  —m*  —  0. 

ALGEBRAIC  PROBLEMS  29 

Solve  the  following  equations  : 

36.    r^  +  3/=ll,  39.    Vx+V2/  =  5, 

36.    x'-y^f-^x,  4Q    x-hy  =  13, 

^  +  y  =  Kx-fy  ^_^^_^g^ 

«^-   ^  +  3/  =  10,  ,,     a-  +  a:y  +  2r'  =  39, 

yVa;=12.  ar^4-a:z  +  z2  =  i9, 

38.  x^4-r  =  13,  y2^y2-f-z*  =  49. 
y  +  X2/  =  9. 

42.   5t/(a;«  +  l)-3ar'(2/«+l)=0, 
15y«(ar^4-l)-x(/  +  l)=0. 

43.  A  farmer  being  asked  how  many  acres  he  had,  replied, 
"  My  land  is  square.  I  have  plowed  just  2  rods  wide  around, 
and  have  plowed  just  \  my  land."     How  many  acres  has  he  ? 

44.  From  a  10-gallon  keg  of  wine  a  man  filled  a  jug.  He 
then  filled  the  keg  with  water,  and  repeated  the  operation  a 
second  time,  when  he  found  the  keg  contained  equal  amounts 
of  water  and  wine.     Find  the  capacity  of  the  jug. 

45.  If  a  certain  number  is  divided  by  32,  the  remainder  is 
25 ;  if  divided  by  25,  the  remainder  is  19 ;  and  if  divided  by 
19,  the  remainder  is  11.     What  is  the  number? 

46.  If  Dr.  A  loses  3  patients  out  of  7 ;  Dr.  B,  4  out  of  13 ; 
and  Dr.  C,  5  out  of  19 ;  what  chance  has  a  sick  man  for  his 
life,  who  is  dosed  by  the  three  doctors  for  the  same  disease  ? 

47.  Said  Robin  to  Richard,  "  If  ever  I  come 

To  the  age  that  you  are,  brother  mine, 
Our  ages  united  would  amount  to  the  sum 

Of  years  making  ninety-nine." 
Said  Richard  to  Robin,  "  That's  certain,  and  if  it  be  fair 

For  us  to  look  forward  so  far, 
I  then  shall  be  double  the  age  that  you  were, 

When  I  was  the  age  that  you  are." 


30  MATHEMATICAL  WRINKLES 

48.  A  tells  the  truth  2  times  out  of  3,  B  6  times  out  of  7, 
and  C  4  times  out  of  5.  What  is  the  probability  of  the  truth 
of  an  assertion  that  A  and  B  affirm  and  C  denies  ? 

49.  A  plank  16  feet  long  with  a  weight  of  196  pounds,  on 
one  end  balances  across  a  fulcrum  placed  1  foot  from  the 
196-pound  weight.     What  is  the  weight  of  the  plank  ? 

50.  A  man  desires  to  purchase  eggs  at  5  cents,  1  cent,  and 
■i-  cent,  respectively,  in  such  numbers  that  he  will  obtain  100 
eggs  for  a  dollar.     How  many  solutions  in  rational  integers  ? 

51.  Ann's  brother  started  to  school.  On  the  first  day  the 
teacher  asked  him  his  age.  He  replied,  "  When  I  was  born, 
Ann  was  ^  the  age  of  mother  and  is  now  ^  as  old  as  father, 
and  I  am  J  of  mother's  age.  In  4  years  I  shall  be  i  as  old 
as  father."     How  old  is  Ann's  brother  ? 

52.  Solve  for  x : 

Om+6      Qm-1      Q(n+1)* 

^6  ^  _J_ !_» i.y ^  ,21^-^-n  .  32-1-^-1). 

53.  My  wife  was  born  ,, 

lL(f)UL  2^^'      2«-ijL(2  .  3  .  3K  e^ijl 

What  was  her  age  August  10,  1904  ? 

Note. — Problems  54-67,  inclusive,  are  from  Bowser's  "College  Al- 
gebra." 

54.  Express  with  positive  exponents 


■V(a  +  &)'x(a  +  6)-t. 
55.   Extract  the  square  root  of 

6-f.2V2  +  2V3-|-2V6. 
66.    Extract  the  square  root  of 

54.V10- V6- Vi5. 

57.    Solve  a;~^  +  a;~^  =  6. 


xy  =2*. 

61. 

x^  +  y^  =  14a^/, 

x-{-y  =  a. 

62. 

m'<^)'-^- 

xy  —  x  —  y  —  bAi. 

ALGEBRAIC  PROBLEMS  31 

58.    Solve  a;^-|-x^  =  1056. 

69.   Solve  -Jt—     {a^-h^)x= -^ -. 

60.   Solve  the  following : 

6(a^  +  y*  +  2^)  =  13(0;  ^- 2/ -h  2)  =  *fL, 


63.   a?  +  y{xy-l)  =  0, 
f-x(xy  +  l)  =  0. 


65.    (x^-hl)y  =  (f-¥l)^y 
{f^l)x=9(x^  +  l)f, 

66.  A  offers  to  run  three  times  round  a  course  while  B  runs 
twice  round,  but  A  gets  only  150  yards  of  his  third  round  fin- 
ished when  B  wins.  A  then  offers  to  run  four  times  round  to  B 
three  times,  and  now  quickens  his  pace  so  that  he  runs  4  yards 
in  the  time  he  formerly  ran  3  yards.  B  also  quickens  his  so 
that  he  runs  9  yards  in  the  time  he  formerly  ran  8  yards,  but 
in  the  second  round  falls  off  to  his  original  pace  in  the  first 
race,  and  in  the  third  round  goes  only  9  yards  for  10  he  went 
in  the  first  race,  and  accordingly  this  time  A  wins  by  180 
yards.     Determine  the  length  of  the  course. 

67.  On  the  ground  are  placed  n  stones ;  the  distance  between 
the  first  and  second  is  1  yard,  between  the  second  and  third 
3  yards,  between  the  third  and  fourth  5  yards,  and  so  on. 
How  far  will  a  person  have  to  travel  who  shall  bring  them 
one  by  one  to  a  basket  placed  at  the  first  stone  ? 

68.  Sionius  and  his  wife  Lionius  sip  from  the  same  bowl 
filled  with  milk.  Lionius  sips  during  f  of  the  time  which 
Sionius  would  take  to  empty  the  bowl ;  then  Lionius  stops  and 


32  MATHEMATICAL    WRINKLES 

hands  it  to  Sionius  to  finish.  If  both  had  sipped  together,  the 
bowl  would  have  been  emptied  6  minutes  sooner,  and  Lionius 
would  have  received  |  of  the  milk  which  Sionius  sipped  after 
receiving  the  bowl  from  Lionius.  In  what  time  would  Sionius 
and  Lionius  sipping  together  empty  the  bowl  ? 

69.  Once,  in  classic  days,  Silenus  lay  asleep,  a  goatskin 
filled  with  wine  near  him.  Dionysius  passing  by,  profited  by 
seizing  the  skin,  and  drinking  for  |  of  that  time  in  which 
Silenus  alone  could  have  emptied  said  skin.  At  this  point  Si- 
lenus awoke,  and  seeing  what  was  happening,  snatched  away 
the  precious  skin,  and  finished  it. 

Now,  had  both  started  together,  and  drunk  simultaneously, 
they  would  have  consumed  the  wine  skin  in  2  hours  less 
time.  And,  in  this  case,  Dionysius'  share  would  have  been 
J  as  much  as  Silenus  did  secure,  by  waking  and  snatching  the 
skin.  In  what  time  would  either  one  of  them  alone  finish  the 
goatskin  ? 

70.  Three  regiments  move  north  as  follows :  B  is  20  miles 
east  of  A ;  C  is  20  miles  south  of  B,  and  each  marches  20 
miles  between  the  hours  of  5  a.m.  and  3  p.m.  A  horseman 
with  a  message  from  C  starts  at  5  a.m.  and  rides  north  till  he 
overtakes  B,  then  sets  a  straight  course  for  the  point  at  which 
he  calculates  to  overtake  A,  then  sets  a  straight  course  for  the 
next  point  at  which  he  will  again  overtake  B,  then  rides  south 
to  the  point  where  he  first  overtook  B,  reaching  that  point  at 
the  same  time  as  C,  namely,  3  p.m.  What  uniform  rate  of 
travel  enabled  the  messenger  to  do  this  ? 

71.  Three  men  and  a  boy  agree  to  gather  the  apples  in  an 
orchard  for  $  50.  The  boy  can  shake  the  apples  in  the  same 
time  that  the  men  can  pick  them,  but  any  one  of  the  men  can 
shake  them  25  %  faster  than  the  other  two  men  and  boy  can 
pick  them.     Find  the  amount  due  each. 


GEOMETRICAL  EXERCISES 

1.  Construct  a  trapezoid  having  given  the  sum  of  the 
parallel  sides,  the  sum  of  the  diagonals,  and  the  angle  formed 
by  the  diagonals. 

2.  If  three  equal  circles  are  tangent  to  each  other,  each  to 
each,  and  inclose  a  space  between  the  three  arcs  equal  to  200 
square  feet,  find  the  diameter  of  each  circle. 

3.  An  iron  rod  of  a  certain  length  stands  against  the  side 
of  a  house ;  if  it  is  pulled  out  4  feet  at  the  bottom,  the  top 
moves  down  the  side  of  the  house  a  distance  equal  to  ^  the 
rod.     Find  the  length  of  the  rod. 

4.  A  circle  whose  area  is  1809.561  square  feet  is  described 
upon  the  perpendicular  of  a  right  triangle  as  a  diameter. 
From  the  point  where  the  circumference  cuts  the  hypotenuse 
a  tangent  to  the  circle  is  drawn,  which  cuts  the  base.  If  the 
shortest  distance  from  the  point  of  intersection  of  the  tangent 
with  the  base  to  the  perpendicular  is  18  feet,  what  is  the  length 
of  the  hypotenuse? 

5.  The  number  of  cubic  inches  contained  by  two  equal 
opposite  spherical  segments,  together  with  the  number  of 
cubic  inches  contained  by  the  cylinder  included  between  these 
segments,  is  600.  If  this  be  J  of  the  number  of  cubic  inches 
contained  by  the  whole  sphere,  find  the  height  of  the  cylinder. 

6.  The  sum  of  the  sides  of  a  right-angled  triangle  is  200 
feet.  What  is  its  area,  the  hypotenuse  being  4  times  the  per- 
pendicular let  fall  upon  it  from  the  right  angle  ? 

33 


34  MATHEMATICAL   WRINKLES 

7.  In  a  right-angled  triangle  the  hypotenuse  is  100  feet, 
and  a  line  bisecting  the  right  angle  and  terminating  in  the 
hypotenuse  is  14.142  feet.  Eind  the  length  of  each  of  the 
other  two  sides. 

8.  Two  posts,  one  of  which  is  24,  and  the  other  16  feet 
high  are  100  feet  apart.  What  is  the  length  of  a  rope  just 
long  enough  to  touch  the  ground  between  them,  the  ends  of 
the  rope  being  fastened  to  the  top  of  each  post? 

9.  A  ladder  30  feet  long  leans  against  a  perpendicular  wall 
at  an  angle  of  30°.  How  far  will  its  middle  point  move,  pro- 
vided the  top  moves  down  the  wall  until  it  reaches  the  ground  ? 

10.  A  man  owns  a  piece  of  land  in  the  form  of  a  right- 
angled  triangle.  The  sum  of  the  sides  about  the  right  angle  is 
70  feet  and  their  difference  equals  the  length  of  a  line  parallel 
to  the  shorter  side,  dividing  the  triangle  into  two  equal  parts. 
Determine  the  length  of  the  shorter  side. 

11.  Required  the  greatest  right  triangle  which  can  be  con- 
structed upon  a  given  line  as  hypotenuse. 

12.  A  man  has  a  lot  the  shape  of  which  is  an  equilateral 
triangle,  with  an  area  of  60  square  rods.  How  long  a  rope 
will  be  required  to  graze  his  horse  over  ^  the  lot,  provided  he 
ties  the  rope  to  a  corner  post? 

13.  An  iron  ball  3  inches  in  diameter  weighs  8  pounds. 
Eind  the  weight  of  an  iron  shell  3  inches  thick,  whose  external 
diameter  is  30  inches. 

14.  Eind  the  altitude  of  the  maximum  cylinder  that  can  be 
inscribed  in  a  cone  whose  altitude  is  9  feet  and  whose  base  is 
6  feet. 

15.  Construct  a  plane  triangle  having  given  the  base,  the 
vertical  angle,  and  the  bisector  of  the  vertical  angle. 

16.  How  much  of  the  earth's  surface  would  a  man  see  if  he 
WQre  rg^ised,  to  the  height  of  the  diameter  above  it  ? 


GEOMETRICAL  EXERCISES  35 

17.  To  what  height  must  a  man  be  raised  above  the  earth 
in  order  that  he  may  see  \  of  its  surface  ? 

18.  What  part  of  the  surface  of  a  sphere  20  feet  in  diameter 
is  illuminated  by  a  lamp  100  feet  from  the  surface  of  the 
sphere  ? 

19.  If  the  earth  is  assumed  to  be  a  sphere  of  4000  miles 
radius,  how  far  at  sea  can  a  lighthouse  110  feet  high  be  seen  ? 

20.  Determine  the  sides  of  an  equilateral  triangle,  having 
given  the  lengths  of  the  three  perpendiculars  drawn  from  any 
point  within  to  the  sides. 

21.  Find  the  number  of  cubic  inches  of  water  that  a  bowl 
will  hold,  whose  shape  is  that  of  a  spherical  segment,  10  inches 
in  height,  the  diameter  of  the  top  being  40  inches. 

22.  Find  the  side  of  the  lai-gest  cube  that  can  be  cut  from 
a  globe  24  inches  in  diameter. 

23.  Which  is  the  greater  —  3  solid  inches,  or  3  inches  solid  ? 

24.  Three  men  living  60  miles  from  one  another  wish  to  dig 
a  well  that  will  be  the  same  distance  from  each  of  their  homes. 
Where  must  they  dig  the  well  ? 

25.  Bisect  a  given  quadrilateral  by  a  straight  line  drawn 
through  a  vertex. 

26.  One  arm  of  a  right  triangle  is  30  feet  and  the  perpen- 
dicular from  the  vertex  of  the  right  triangle  to  the  hypotenuse 
is  24  feet.     Find  the  area  of  the  triangle. 

27.  Three  chords,  lengths  6,  8,  and  10,  just  go  around  in  a 
semicircle.     Find  the  radius  of  the  circle. 

28.  A  cone,  a  half  globe,  and  a  cylinder,  of  the  same  base 
and  altitude,  are  as  1 :  2  :  3. 

29.  Two  sides  of  a  triangle  are  3  feet  and  8  feet,  respec- 
tively, and  inclose  an  angle  of  60°.     Find  the  third  side. 


36  MATHEMATICAL  WRINKLES 

30.  A  rectangular  garden  is  40  feet  by  60  feet.  It  is  sur- 
rounded by  a  road  of  uniform  width,  the  area  of  which  is 
equal  to  the  area  of  the  field.     Find  the  width  of  the  road. 

31.  The  sum  of  the  two  crescents  made  by  describing  semi- 
circles outward  on  the  two  sides  of  a  right  triangle  and  a  semi- 
circle toward  them  on  the  hypotenuse,  is  equivalent  to  the 
right  triangle. 

32.  Prove  that  the  circle  through  the  middle  points  of  the 
sides  of  a  triangle  passes  through  the  feet  of  the  perpendicu- 
lars from  the  opposite  vertices,  and  through  the  middle  points 
of  the  segments  of  the  perpendiculars  included  between  their 
point  of  intersection  and  the  vertices. 

33.  What  is  the  volume  of  the  frustum  of  a  sphere,  the 
radius  of  whose  upper  base  is  3  feet  and  lower  base  4  feet,  and 
altitude  1  foot  ? 

34.  If  a  circle  rolls  on  the  inside  of  a  fixed  circle  of  double 
the  radius,  find  the  length  of  the  path  that  any  fixed  point  in 
the  circumference  of  the  moving  circle  will  trace  out. 

35.  Find  the  diameter  of  a  circle  inscribed  in  a  triangle 
whose  sides  are  6,  8,  and  10  feet,  respectively. 

36.  Find  the  diameter  of  a  circle  circumscribed  about  a 
triangle  whose  sides  are  6,  8,  and  10  feet,  respectively. 

37.  What  is  the  area  of  an  equilateral  triangle  whose  sides 
are  100  inches  ? 

38.  What  is  the  area  of  a  tetragon  (square)  whose  sides  are 
100  inches  ? 

39.  What  is  the  area  of  a  regular  pentagon  whose  sides  are 
100  inches  ? 

40.  What  is  the  area  of  a  regular  hexagon  whose  sides  are 
10  feet? 


GEOMETRICAL  EXERCISES  37 

41.  What  is  the  area  of  a  regular  heptagon  whose  sides  are 
10  feet  ? 

42.  What  is  the  area  of  a  regular  octagon  whose  sides  are 
10  feet  ? 

43.  What  is  the  area  of  a  regular  nonagon  whose  sides  are 
10  feet  ? 

44.  What  is  the  area  of  a  regular  decagon  whose  sides  are 
10  feet  ? 

45.  What  is  the  area  of  a  regular  undecagon  whose  sides 
are  10  feet? 

46.  What  is  the  area  of  a  regular  dodecagon  whose  sides 
are  10  feet? 

47.  Find  the  side  of  an  inscribed  square  of  a  triangle  whose 
base  is  10  feet  and  altitude  4  feet. 

48.  Find  the  diameter  of  a  circle  of  which  the  height  of  an 
arc  is  6  inches  and  the  chord  of  half  the  arc  is  10  inches. 

49.  Find  the  height  of  an  arc,  when  the  chord  of  the  arc  is 
10  inches  and  the  radius  of  the  circle  is  8  inches. 

50.  Find  the  chord  of  half  an  arc,  when  the  chord  of  the 
arc  is  20  feet  and  the  height  of  the  arc  is  2  feet. 

51.  Find  the  chord  of  half  an  arc,  when  the  chord  of  the 
arc  is  10  inches  and  the  radius  of  the  circle  is  8  inches. 

52.  Find  the  side  of  a  circumscribed  polygon,  when  the  side 
of  a  similar  inscribed  polygon  is  10  feet  and  the  radius  of  the 
circle  is  30  feet. 

53.  A  log  10  feet  long,  2  feet  in  diameter  at  one  end  and 
3  feet  at  the  other,  is  rolled  along  till  the  larger  end  describes 
a  circle.     Find  the  diameter  of  the  circle. 

54.  At  the  extremities  of  the  diameter  of  a  circular  park 
stand  two  electric  light  posts,  one  12  feet  high  and  the  other 
18  feet  high.     What  points  on  the  circumference  of  the  park 


38  MATHEMATICAL  WEINKLES 

are  equidistant  from  the  tops  of  the  posts,  the  diameter  of  the 
park  being  100  feet  ? 

55.  What  is  the  circumference  of  the  largest  circular  ring 
that  can  be  put  in  a  cubical  box  whose  edge  is  4  feet  ? 

56.  What  is  the  side  of  the  largest  square  that  can  be  in- 
scribed in  a  semicircle  whose  diameter  is  2Vo  feet? 

57.  What  is  the  volume  of  the  largest  cube  that  can  be 
inscribed  in  a  hemisphere  whose  diameter  is  3  feet  ? 

58.  In  a  triangle  whose  base  is  30  inches  and  altitude  18 
inches  a  square  is  inscribed.     Find  its  area. 

59.  Two  equal  circles  of  10-inch  radii  are  described  so  that 
the  center  of  each  is  on  the  circumference  of  the  other.  Find 
the  area  of  the  curvilinear  figure  intercepted  between  the  two 
circumferences. 

60.  Two  equal  circles  of  8-inch  radii  intersect  so  that 
their  common  chord  is  equal  to  their  radius.  Find  the  area 
of  the  curvilinear  figure  intercepted  between  the  two  cir- 
cumferences. 

61.  Find  the  area  of  a  zone  whose  altitude  is  4  feet  on  a 
sphere  whose  radius  is  10  feet. 

62.  Find  the  volume  of  a  segment  of  a  sphere  whose  altitude 
is  1  foot  and  the  radius  of  the  base  2  feet. 

63.  Mr.  Brown  has  a  plank  of  uniform  thickness  10  feet 
long,  12  inches  wide  at  one  end  and  5  inches  at  the  other.  How 
far  from  the  large  end  must  it  be  cut  straight  across  so  that  the 
two  parts  shall  be  equal  ? 

64.  Having  given  the  lesser  segment  of  a  straight  line 
divided  in  extreme  and  mean  ratio,  to  construct  the  whole  line. 

65.  Find  the  volume  of  a  spherical  shell  whose  two  surfaces 
are  64  tt  and  36  tt. 

66.  To  construct  a  triangle  having  given  the  three  medians. 


GEOMETRICAL  EXERCISES  39 

67.  Two  sides  of  a  quadrilateral  lot  run  east  216  feet  and 
north  63  feet.  If  the  other  two  sides  measure  135  and  180  feet, 
respectively,  what  is  its  area  in  square  yards  ? 

68.  If  the  perimeter  of  a  right  triangle  is  240  rods  and  the 
radius  of  the  inscribed  circle  20  rods,  what  are  the  sides  ? 

69.  On  a  hillside  which  slopes  11  feet  in  61  feet  of  its 
length,  stands  an  upright  pole.  If  this  pole  should  break  at 
a  certain  point  and  fall  up  hill,  the  top  would  strike  the 
ground  61  feet  from  the  base  of  the  pole ;  but  if  it  should  fall 
down  hill,  its  top  would  strike  the  ground  4S^  feet  from  the 
base  of  the  pole.     Find  the  length  of  the  pole. 

70.  A  house  and  barn  are 
25  rods  apart.  The  house 
is  12  rods  and  the  barn  5 
rods  from  a  brook  running 
in  a  straight  line.  What  is 
the  shortest  distance  one 
must  walk  from  the  house 

to  get  a  pail  of  water  from  the  brook  and  carry  it  to  the  barn  ? 

71.  Construct  geometrically  the  square  root  of  any  number,  n. 

72.  Construct  a  triangle  having  given  the  base,  the  median 
upon  the  base,  and  the  difference  between  the  base  angles. 

73.  A  man  owning  a  rectangular  field  300  feet  by  600  feet, 
wishes  to  lay  out  driveways  of  equal  width  having  the  diago- 
nals of  the  field  as  center  lines,  and  such  that  the  area  of  the 
driveways  shall  be  J  of  the  area  of  the  field.  Determine  the 
width  of  the  driveways. 

74.  Two  ladders  14  feet  apart  at  their  base  touch  each 
other  at  the  top.  Each  is  inclined  the  same,  and  a  round 
10  feet  up  on  either  side  is  as  far  from  the  top  as  it  is 
from  the  base  of  the  other  ladder.  Get  the  length  of  the 
ladders. 


uouse^ 

""""'•^ 

Bam 

1 

Brook 

iO 

40  MATHEMATICAL   WRINKLES 

75.  A  tree  123  feet  high  breaks  off  a  certain  distance  up, 
and  the  moment  the  top  strikes  a  stump  15  feet  high  the 
broken  part  points  to  a  spot  108  feet  from  the  base  of  the 
tree.     Find  the  length  of  the  part  broken  off. 

76.  Divide  a  triangle  into  three  equivalent  parts  by  lines 
drawn  from  a  point  P  within  the  triangle. 

77.  From  a  point  P  without  a  circumference,  to  draw  a 
secant  which  is  bisected  by  the  circumference. 

78.  To  construct  a  triangle  having  given  the  three  feet  of 
the  altitudes. 

79.  If  from  any  point  in  the  circumference  of  a  circle  per- 
pendiculars be  dropped  upon  the  sides  of  an  inscribed  triangle 
(produced,  if  necessary),  the  feet  of  the  perpendiculars  are  in 
a  straight  line. 

80.  Inside  a  square  10-acre  lot  a  cow  was  tethered  to  the 
fence  at  a  point  1  rod  from  the  corner  by  a  rope  just  long 
enough  to  allow  her  to  graze  over  an  acre  of  ground.  How 
long  was  the  rope  ? 

81.  From  any  point  P  in  the  bisector  of  the  angle  A  in 
a  triangle  ABCy  perpendiculars  PA\  PB\  PC  are  drawn  to 
the  three  sides.  Prove  PA'  and  JB'C"  intersect  in  the  median 
from  A. 

82.  If  the  bisectors  of  two  angles  of  a  triangle  are  equal,  the 
triangle  is  isosceles. 

83.  In  a  right  triangle  the  bisector  of  the  right  angle  also 
bisects  the  angle  between  the  perpendicular  and  the  median 
from  the  vertex  of  the  right  angle  to  the  hypotenuse. 

84.  Find  the  locus  of  a  point  the  sum  or  the  difference  of 
whose  distances  from  two  fixed  straight  lines  is  given. 

85.  The  bisector  of  an  angle  of  a  triangle  is  less  than  half 
the  sum  of  the  sides  containing  the  angle. 


GEOMETRICAL   EXERCISES  41 

86.  The  difference  between  the  acute  angles  of  a  right  triangle 
is  equal  to  the  angle  between  the  median  and  the  perpendicu- 
lar drawn  from  the  vertex  of  the  right  angle  to  the  hypotenuse. 

87.  A  hollow  rubber  ball  is  2  inches  in  diameter  and  the 
rubber  is  -^jr  inch  thick.  How  much  rubber  would  be  used 
in  the  manufacture  of  1000  such  balls  ? 

88.  Having  given  two  concentric  circles,  draw  a  chord  of  the 
larger  circle,  which  shall  be  divided  into  three  equal  parts  by 
the  circumference  of  the  smaller  circle. 

89.  The  distances  from  a  point  to  the  three  nearest  corners 
of  a  square  are  1  inch,  2  inches,  and  2J  inches.  Construct  the 
square. 

90.  Draw  a  chord  of  given  length  through  a  given  point, 
within  or  without  a  given  circle. 

91.  Find  the  greatest  segment  of  a  line  10  inches  long,  when 
it  is  divided  in  extreme  and  mean  ratio. 

92.  In  a  quadrilateral  ABCD,  AB  =  10,  BC  =  17,  CD  =  13, 
DA  =  20,  and  AC  =  21.     Find  the  diagonal  BD. 

93.  To  divide  a  trapezoid  into  two  similar  trapezoids  by  a 
line  parallel  to  the  base. 

94.  From  a  given  point  in  a  circumference,  to  draw  a  chord 
that  is  bisected  by  a  given  chord. 

95.  In  a  given  line  AB,  to  find  a  point  C  such  that  AC:  BC 

=  1  :  V2: 

96.  From  a  given  rectangle  to  cut  off  a  similar  rectangle  by 
a  line  parallel  to  one  of  its  sides. 

97.  Find  the  locus  of  a  point  in  space  the  ratio  of  whose 
distances  from  two  given  points  is  constant. 

98.  Find  the  locus  of  a  point  whose  distance  from  a  fixed 
straight  line  is  in  a  given  ratio  to  its  distance  from  a  fixed 
plane  perpendicular  to  that  line. 


42  MATHEMATICAL   WEINKLES 

99.    Any  point  in  the  bisector  of  a  spherical  angle  is  equally 
distant  from  the  sides  of  the  angle. 

100.  If  any  number  of  lines  in  space  meet  in  a  point,  the  feet 
of  the  perpendiculars  drawn  to  these  lines  from  another  point 
lie  on  the  surface  of  a  sphere. 

101.  If  the  angles  at  the  vertex  of  a  triangular  pyramid  are 
right  angles,  and  the  lateral  edges  are  equal,  prove  that  the 
sum  of  the  perpendiculars  on  the  lateral  faces  from  any  point 
in  the  base  is  constant. 

102.  A  plane  bisecting  two  opposite  edges  of  a  regular 
tetraedron  divides  the  tetraedron  into  two  equal  polyedrons. 

103.  The  volume  of  a  truncated  triangular  prism  is  equal  to 
the  product  of  the  lower  base  by  the  perpendicular  on  the 
lower  base  from  the  intersection  of  the  medians  of  the  upper 
base. 

104.  The  point  of  intersection  of  the  perpendiculars  erected 
at  the  middle  of  each  side  of  a  triangle,  the  point  of  intersec- 
tion of  the  three  medians,  and  the  point  of  intersection  of  the 
three  perpendiculars  from  the  vertices  to  the  opposite  sides  are 
in  a  straight  line ;  and  the  distance  of  the  first  point  from  the 
second  is  half  the  distance  of  the  second  from  the  third. 

105.  Three  circles  are  tangent  externally  at  the  points  A, 
B,  and  C,  and  the  chords  AB  and  AC  are  produced  to  cut  the 
circle  BC  at  D  and  E.     Prove  that  DE  is  a  diameter. 

106.  A  cylindrical  bucket  without  a  top  is  6  inches  in  cir- 
cumference and  4  inches  high.  On  the  inside  of  the  vessel 
1  inch  from  the  top  is  a  drop  of  honey,  and  on  the  opposite  side 
of  the  vessel  1  inch  from  the  bottom,  on  the  outside,  is  a  fly. 
How  far  will  the  fly  have  to  go  to  reach  the  honey  ? 

107.  P  is  any  point  on  the  circumcircle  of  an  equilateral 
triangle  ABC;  AP,  BP  meet  BC,  CA  respectively  in  X,  Y. 
Prove  BX  -  AY  is  constant. 


GEOMETRICAL  EXERCISES  43 

108.  Find  the  locus  of  all  points  from  which  two  unequal 
circles  subtend  equal  angles. 

109.  Show  that  any  two  perpendicular  lines  terminated  by 
the  opposite  sides  of  a  square  are  equal  to  one  another,  and  by 
this  property  show  how  to  escribe  a  square  to  a  given  quadri- 
lateral. 

110.  If  the  incircle  passes  through  the  centroid  of  the  tri- 
angle, find  the  relation  between  the  sides  a,  6,  and  c. 

111.  If  through  a  point  O  within  a  triangle  ^BC  parallels 
EFy  GHy  IK  to  the  sides  be  drawn,  the  sum  of  the  rectangles 
of  their  segments  is  equal  to  the  rectangle  contained  by  the 
segments  of  any  chord  of  the  circumscribing  circle  passing 
through  0. 

112.  If  two  chords  intersect  at  right  angles  within  a  circle, 
the  sum  of  the  squares  on  their  segments  equals  the  square  on 
the  diameter. 

113.  If  from  a  point  A^  without  a  circle,  two  secants,  ACD 
and  AGKy  are  drawn,  the  chords  C/iTand  DG  intersect  on  the 
chord  of  contact  of  the  tangents  from  the  point  A  to  the  circle. 

114.  If  from  a  given  point  without  a  given  circle  any  num- 
ber of  secants  are  drawn,  the  chords  joining  the  points  of 
intersection  of  the  secants  with  the  circle  all  cross  on  the  same 
straight  line. 

115.  To  draw  a  tangent  from  a  given  external  point  to  a 
given  circle  by  means  of  a  ruler  only. 

116.  Of  all  polygons  constructed  with  the  same  given  sides, 
the  cyclic  polygon  is  the  maximum. 

117.  The  square  on  the  side  of  a  regular  inscribed  pentagon 
is  equal  to  the  square  on  the  side  of  a  regular  inscribed  hexa- 
gon, plus  the  square  on  the  side  of  a  regular  inscribed  decagon. 

118.  The  area  of  an  inscribed  regular  dodecagon  is  three 
times  the  square  of  the  radius  of  the  circle. 


44  MATHEMATICAL   WRINKLES   - 

119.  The  square  of  the  side  of  an  inscribed  equilateral  tri- 
angle is  equal  to  the  sum  of  the  squares  of  the  sides  of  an 
inscribed  square  and  inscribed  regular  hexagon. 

120.  Construct  a  circumference  equal  to  three  times  a  given 
circumference. 

121.  Construct  a  circle  equivalent  to  three  times  a  given 
circle. 

122.  If  ABCD  be  a  cyclic  quadrilateral,  and  if  we  describe 
any  circle  passing  through  the  points  A  and  B,  another  through 
B  and  C,  a  third  through  G  and  D,  and  a  fourth  through  D 
and  A  ;  these  circles  intersect  successively  in  four  other  points, 
E,  F,  G,  H,  forming  another  cyclic  quadrilateral. 

123.  Construct  a  triangle,  given  the  altitude,  the  median, 
and  the  angle  bisector,  all  from  the  same  vertex. 

124.  Prove  that  the  circumcircle  of  a  triangle  bisects  each 
of  the  six  segments  determined  by  the  incenter  and  the  three 
excenters  of  the  triangle. 

125.  If  A,  B,  G  are  three  collinear  points,  and  if  K  is  any 
other  point,  prove  that  the  circumcenters  of  the  triangles  KBG, 
KCA,  and  KAB  are  concyclic  with  K. 

126.  If  the  diameter  of  a  circle  be  divided  into  any  number 

of  segments,  and  circumferences  be  de- 
scribed upon  these  segments  as  diameters, 
the  sum  of  these  circumferences  is  equal  to 
the  circumference  of  the  original  circle. 


127.  I  own  a  square  garden  as  shown  in 
the  above  diagram.  Within  the  garden 
stands  a  tree  30  feet,  40  feet,  and  50  feet 

respectively  from  three  successive  corners.     How  much  land 

have  1  ? 


GEOMETRICAL   EXERCISES 


45 


The  Famous  Nine-Point  Circle. 

128.  (a)  If  a  circle  be  described  about  the  pedal  triangle  of 
any  triangle,  it  will  pass  through  the  middle  points  of  the 
lints  drawn  from  the  orthocenter  to  the  vertices  of  the  triangle, 
and  through  the  middle  points  of  the  sides  of  the  triangle,  in 
all,  through  nine  points. 

(6)  The  center  of  the  nine-point  circle  is  the  middle  point  of 
the  line  joining  the  orthocenter  and  the  center  of  the  circum- 
circle  of  the  triangle. 

(c)  The  radius  of  the  nine-point  circle  is  half  the  radius  of 
the  circumcircle  of  the  triangle. 

(d)  The  centroid  of  the  triangle  also  lies  on  the  line  join- 
ing the  orthocenter 

and  the  center  of  yl 

the  circumcircle  of        \  /  ; 

the    triangle,    and        \  /       I 

divides    it    in   the  \  X  j 

ratio  of  2:1.  \  ^,^^^,^  j 

(e)  The  sides  of  \  --    -      ^ 
the  pedal  triangle 
intersect  the  sides 
of   the    given    tri-      /^ 
angle  in  the   radi-     i     a'< 
cal  axis  of  the  cir-      \ 
cumscribing       and 
nine-point  circles. 

(/)    The    nine- 
point  circle  is  tan- 
gent    to    the     in-  \  / 
scribed     and      es-                                "'v                       / 

cribed    circles    of  \  ^-- ^-^ 

the  triangle. 

Let  ABD  be  any  triangle,  A\  B\  D\  the  projections  of  the 
vertices  on  the  opposite  sides;  //,  «/,  K,  the  mid-points  of  OAy 


/ 


\ 


•Y  --' 

>> 

A 

II 

^-, 

O 

^ 

/               \ 

; 

K                     1 

46  MATHEMATICAL  WRINKLES 

OB,  OD,  respectively,  0  being  the  orthocenter.  Let  L,  M,  N" 
be  the  mid-points  of  the  sides.  Join  F,  E,  and  D.  The  A  A'B'D' 
is  called  the  pedal  triangle.  The  nine  points  A',  N,  K,  D',  H,  B', 
J,  L,  M  are  concyclic ;  and  the  circle  through  them  is  the  nine- 
point  circle  of  the  triangle. 

For  the  proofs  of  these  theorems,  see  "Finkel's  Mathematical 
Solution  Book  "  and  the  monograph,  "  Some  Noteworthy  Prop- 
erties of  the  Triangle  and  Its  Circles,"  by  Dr.  W.  H.  Bruce, 
president  of  the  North  Texas  State  Normal  School,  Denton. 

129.  If  from  any  point  in  either  side  of  a  right  triangle,  a 
line  is  drawn  perpendicular  to  the  hypotenuse,  the  product  of 
the  segments  of  the  hypotenuse  is  equal  to  the  product  of  the 
segments  of  the  side  plus  the  square  of  the  perpendicular. 

130.  A,  B,  and  C  are  fixed  points.  Describe  a  square  with 
one  vertex  at  A,  so  that  the  sides  opposite  to  A  pass  through 
B  and  G. 

131.  If  ABCD  is  a  cyclic  quadrilateral,  prove  that  the  cen- 
ters of  the  circles  inscribed  in  triangles  ABC,  BCD,  CDA, 
DAB  are  the  vertices  of  a  rectangle. 

132.  A  round  hole  one  foot  in  diameter  is  cut  through  a 
sphere  20  inches  in  diameter.  Find  the  volume  of  the  part 
remaining,  the  axes  of  the  hole  passing  through  the  center  of 
the  sphere. 

133.  Given  the  incenter,  circumcenter,  and  one  excenter  of 
a  triangle,  construct  it. 

134.  Divide  the  triangle  whose  sides  are  7,  15,  20  into  two 
equivalent  parts  by  a  radius  of  the  circumcircle. 

135.  Construct  a  triangle,  given  its  altitude  and  the  radii  of 
the  inscribed  and  circumscribed  circles. 

136.  In  the  semicircle  ABCD  express  the  diameter  AD  in 
terms  of  the  chords  AB,  BC,  and  CD. 


GEOMETRICAL   EXERCISES  47 

137.  On  one  side  of  an  equilateral  triangle  describe  out- 
wardly a  semicircle.  Trisect  the  arc  and  join  the  points  of 
division  with  the  vertex  of  the  triangle.  Find  the  ratio  of  the 
segments  of  the  diameter. 

138.  If  a,  6,  c  are  the  sides  of  a  triangle,  and  5  (a^  -\-b^-\-c^ 
=  G  {ab  +  bc  -\-  ac),  show  that  the  incircle  passes  through  the 
centroid  of  the  triangle. 

139.  If  through  the  vertices  of  any  inscribed  polygon  tan- 
gents are  drawn  forming  a  circumscribed  polygon,  the  con- 
tinued product  of  the  perpendiculars  from  any  point  in  the 
circle  on  the  sides  of  the  inscribed  polygon  is  equal  to  the  con- 
tinued product  of  the  perpendiculars  from  the  same  point  on 
the  sides  of  the  circumscribed  polygon. 

140.  A  lot  100  feet  long  and  60  feet  wide  has  a  walk  ex- 
tending from  one  corner  halfway  around  it,  and  occupying 
one  third  of  the  area.  Required  the  width  of  the  walk.  A 
geometrical  construction  is  desired. 

141.  Construct  a  triangle,  having  given  the  vertical  angle, 
the  sum  of  the  tlfiree  sides,  and  the  perpendicular. 

142.  Prove  that  the  dihedral  angle  of  a  regular  octahedron  is 
the  supplement  of  the  dihedral  angle  of  a  regular  tetrahedron. 

143.  Given  the  three  diagonals  of  an  inscriptible  quadrilat- 
eral, to  construct  the  quadrilateral. 

144.  Pis  a  point  on  the  minor  arc  AB  of  the  circumcircle  of 
the  regular  hexagon  ABCDEF;  prove  that  PE  +  PD  =  PA 
4-  P5  +  PC  4-  PF. 

145.  In  a  right  triangle  the  hypotenuse  is  17  and  the  diam- 
eter of  the  inscribed  circle  6.  Another  equal  circle  is  described 
touching  the  base  produced  and  the  hypotenuse ;  how  far  apart 
are  the  centers  of  the  two  circles  ? 

146.  Two  equal  circular  discs  are  to  be  cut  out  of  a  rectan- 
gular piece  of  paper,  9  inches  long  and  8  inches  wide.  What 
is  the  greatest  possible  diameter  of  the  discs  ? 


MISCELLANEOUS   PROBLEMS 

1.  A  seed  is  planted.  Suppose  at  the  end  of  2  years  it 
produces  a  seed,  and  one  each  year  thereafter ;  each  of  these 
when  2  years  old  produces  a  seed  yearly.  All  the  seeds 
produced  do  likewise.  How  many  seeds  will  be  produced  in 
20  years  ? 

2.  If  a  4-inch  auger  hole  be  bored  diagonally  through  a  12- 
inch  cube,  what  will  be  the  volume  displaced,  the  axis  of  the 
auger  hole  coinciding  with  the  diagonal  of  the  cube  ? 

3.  I  have  a  circular  orchard  110  yards  in  diameter.  How 
many  trees  can  be  set  in  it  so  that  no  two  shall  be  within  16 
feet  of  each  other,  and  no  tree  within  5  feet  of  the  fence  ? 

4.  What  is  the  convex  surface  and  voluifte  of  a  cylindric 
ungula  whose  least  length  is  5  feet,  greatest  length  13  feet,  the 
radius  of  the  base  being  1^-  feet  ? 

5.  AYhat  is  the  length  of  the  arc  whose  chord  is  16  feet  and 
height  6  feet  ? 

6.  Find  the  area  of  a  sector,  having  given  the  chord  of  the 
arc  equal  to  16  feet,  and  the  height  of  the  arc  equal  to  6  feet. 

7.  What  is  the  area  of  a  segment  whose  base  is  6  feet  and 
height  2  feet  ? 

8.  Find  the  volume  of  an  iron  rod  2  inches  in  diameter  and 
10  feet  from  end  to  end  containing  a  loop  whose  inner  diameter 
is  4  inches. 

9.  What  is  the  area  of  a  circular  zone,  one  side  of  which 
is  30  inches  and  the  other  40  inches,  and  the  distance  between 
them  10  inches  ? 

48 


MISCELLANEOUS   PROBLEMS  49 

10.  The  shell  of  a  hollow  iron  ball  is  4  inches  thick,  and 
contains  \  of  the  number  of  cubic  inches  in  the  whole  ball. 
Find  the  diameter  of  the  ball. 

11.  A  rope  60  feet  long  wraps  around  two  trees  6  feet  and 
10  feet  in  diameter,  respectively,  and  crosses  between  them. 
Find  the  distance  between  their  centers. 

12.  On  the  tire  of  a  wheel  4  feet  in  diameter  is  a  black 
spot.  How  far  does  the  spot  move  while  the  wheel  makes  4 
revolutions  ? 

13.  A  fly  lights  on  the  spoke  of  a  carriage  wheel  4  feet  in 
diameter,  1  foot  up  from  the  ground.  How  far  will  the  fly 
have  traveled  when  the  wheel  has  made  2  revolutions  on  a 
level  plane? 

14.  An  eagle  and  a  sparrow  are  in  the  air ;  the  eagle  is  100 
feet  above  the  sparrow.  If  the  sparrow  flies  straight  forward 
in  a  horizontal  line,  and  the  eagle  flies  twice  as  fast  directly 
towards  the  sparrow,  how  far  will  each  fly  before  the  sparrow 
is  caught  ? 

15.  A  cow  is  tethered  to  the  corner  of  a  barn  25  feet  square, 
by  a  rope  100  feet  long.  How  many  square  feet  can  she 
graze  ? 

16.  A  solid  cube  weighs  300  pounds.  If  a  power  is  applied 
at  an  angle  of  45°  at  an  upper  edge  of  the  cube,  how  many 
foot  pounds  will  be  required  to  overturn  the  cube  ? 

17.  A  tree  110  feet  high,  standing  by  the  side  of  a  stream 
100  feet  wide,  is  broken  by  a  storm ;  the  fallen  part  is  unde- 
tached  from  the  stump,  and  its  top  rests  10  feet  above  the 
water  and  points  directly  to  the  opposite  shore.  How  high  is 
the  stump  ? 

18.  At  the  edge  of  a  circular  lake  1  acre  in  area  stands  a 
tree.  What  length  of  rope,  tied  to  this  tree,  will  allow  a  horse 
to  graze  upon  \  of  an  acre  ? 


50  MATHEMATICAL  WRINKLES 

19.  A  horse  is  tied  to  a  stake  in  the  circumference  of  a 
6-acre  field.  How  long  must  the  rope  be  to  allow  him  to  graze 
over  just  1  acre  inside  the  field  ? 

20.  What  is  the  longest  piece  of  carpet  3  feet  wide,  cut 
square  at  the  ends,  that  can  be  put  in  a  room  16  feet  by  20 
feet  ? 

21.  The  fore  wheel  and  the  hind  wheel  of  a  carriage  are  12 
feet  and  15  feet  in  circumference,  respectively;  a  rivet  in  the  tire 
of  each  is  observed  to  be  up  when  the  carriage  starts.  How  far 
will  each  rivet  have  moved  when  they  are  next  up  together  ? 

22.  A  log  40  inches  in  diameter  is  to  be  sawed  by  four  men. 
What  part  of  the  diameter  must  each  man  saw  to  do  ^  of  the 
work  ? 

23.  What  is  the  length  of  a  chord  cutting  off  the  fourth 
part  of  a  circle  whose  radius  is  10  feet  ? 

24.  Find  the  length  of  a  chord  cutting  off  the  third  part  of 
a  circle  whose  diameter  is  40  feet. 

25.  A  tree  80  feet  high  was  broken  in  a  storm  so  that  the 
top  struck  the  ground  40  feet  from  the  foot  of  the  tree.  If 
the  tree  remained  in  contact,  what  was  the  length  of  the  path 
through  which  the  top  of  the  tree  passed  in  falling  to  the 
ground  ? 

26.  By  boring  through  the  center  of  a  wooden  ball,  with  an 
auger  4  inches  in  diameter,  i  of  the  solid  contents  of  the  ball 
is  displaced.     Eind  the  diameter  of  the  ball. 

27.  Find  the  diameter  of  an  auger  that  will  displace  i  of 
the  solid  contents  of  a  ball  5  feet  in  diameter,  by  boring 
through  its  center. 

28.  Three  horses  are  tethered  each  to  a  rope  42  feet  in  length 
to  the  corners  of  an  equilateral  triangle  whose  side  is  80  feet. 
Over  how  many  square  feet  can  each  graze,  provided  they  are 
at  no  time  upon  the  same  ground  ? 


MISCELLANEOUS  PROBLEMS  51 

29.  How  many  acres  of  water  can  a  man  see,  standing  on  a 
ship,  with  his  eyes  just  14  feet  above  the  water,  when  there  is 
no  land  in  sight  ? 

30.  In  a  farmer's  pasture  is  located  a  triangular  house,  the 
length  of  each  side  being  10  yards.  The  farmer  wishing  to 
graze  his  horse  finds  that  stakes  are  not  plentiful  and  decides 
to  tie  the  rope  to  one  corner  of  the  house.  If  the  rope  is  long 
enough  to  allow  the  horse  to  graze  30  yards  from  the  corner  of 
the  house,  over  how  much  ground  can  the  horse  graze  ? 

31.  Three  men  wish  to  carry  each  J  of  an  8-foot  log  of  uni- 
form size  and  density.  Where  must  the  hand  stick  be  placed  so 
tliat  the  one  at  the  end  of  the  log  and  the  others  at  the  ends  of 
the  stick  shall  each  carry  equal  weights  ? 

32.  If  three  equal  circles  are  tangent  to  each  other,  each  to 
each,  and  inclose  a  space  between  the  three  arcs  equal  to  100 
square  inches,  find  their  radius. 

33.  If  three  equal  circles  are  tangent  to  each  other,  each  to 
each,  with  a  radius  of  10  inches,  find  the  area  of  the  space 
inclosed  between  the  three  arcs. 

34.  If  4  acres  pasture  40  sheep  4  weeks,  and  8  acres  pasture 
66  sheep  10  weeks,  how  many  sheep  will  20  acres  pasture  50 
weeks,  the  grass  growing  uniformly  all  the  time  ? 

35.  A  rabbit  60  yards  due  east  of  a  hound  is  running  due 
south  20  feet  per  second ;  the  hound  gives  chase  at  the  rate  of 
25  feet  per  second.  How  far  will  each  run  before  the  rabbit 
is  caught  ? 

36.  How  many  fruit  trees  can  be  set  out  upon  a  space  100 
feet  square,  allowing  no  two  to  be  nearer  each  other  than  10  feet  ? 

37.  How  many  stakes  can  be  driven  down  upon  a  space  12 
feet  square,  allowing  no  two  to  be  nearer  each  other  than  1 
foot? 


52  MATHEMATICAL   WRINKLES 

38.  The  sum  of  the  sides  of  a  triangle  is  100.  The  angle  at 
A  is  double  that  at  B,  and  the  angle  at  B  is  double  that  at  C. 
Find  the  sides. 

39.  A  conical  glass  4  inches  in  diameter  and  6  inches  in 
altitude,  is  filled  with  water.  How  much  water  will  run  out  if 
it  be  turned  through  an  angle  of  45°  ? 

40.  At  what  latitude  is  the  circumference  of  a  parallel  half 
that  of  the  equator,  regarding  the  earth  a  perfect  sphere  ? 

41.  The  difference  between  the  circumscribed  and  inscribed 
squares  of  a  circle  is  72.     What  is  the  area  of  the  circle  ? 

42.  A  drawer  made  of  inch  boards  is  8  inches  wide,  6  inches 
deep,  and  slides  horizontally.  How  far  must  it  be  drawn  out 
to  put  into  it  a  book  4  inches  thick,  6  inches  wide,  and  9 
inches  long  ? 

43.  With  what  velocity  must  a  pail  of  water  be  whirled 
over  the  head  to  prevent  the  water  from  falling  out,  the  radius 
of  the  circle  of  revolution  being  4  feet  ? 

44.  Two  hunters  killed  a  deer,  and  wishing  to  ascertain  its 
weight  they  placed  a  rail  across  a  fence  so  that  it  balanced 
with  one  on  each  end.  They  then  exchanged  places,  and  the 
lighter  man  taking  the  deer  in  his  lap,  the  rail  again  balanced. 
Find  the  weight  of  the  deer,  the  hunters'  weights  being  160 
and  200  pounds. 

45.  At  each  corner  of  a  square  pasture  whose  sides  are  100 
feet  a  cow  is  tied  with  a  rope  100  feet  long.  What  is  the  area 
of  the  part  common  to  the  four  cows  ? 

46.  Find  the  volume  generated  by  the  revolution  of  a  circle 
10  feet  in  diameter  about  a  tangent. 

47.  Find  the  volume  generated  by  revolving  a  semicircle 
20  inches  in  diameter  about  a  tangent  parallel  to  its  diam- 
eter. 


MISCELLANEOUS  PROBLEMS  63 

48.  A  circle  of  10  inches  radius,  with  an  inscribed  regular 
hexagon,  revolves  about  an  axis  of  rotation  20  inches  distant 
from  its  center  and  parallel  to  a  side  of  the  hexagon.  Find  the 
difference  in  area  of  the  generated  surfaces. 

49.  Find  the  difference  in  the  volumes  of  the  two  generated 
solids. 

50.  An  equilateral  triangle  rotates  about  an  axis  without  it, 
parallel  to,  and  at  a  distance  10  inches  from  one  of  its  sides. 
Find  the  surface  thus  generated,  a  side  of  the  triangle  being 
4  inches. 

61.  A  rectangle  whose  sides  are  6  inches  and  18  inches  is 
revolved  about  an  axis  through  one  of  its  vertices,  and  parallel 
to  a  diagonal.     Find  the  surface  thus  generated. 

52.  Find  the  surface  of  a  square  ring  described  by  a  square 
foot  revolving  round  an  axis  parallel  to  one  of  its  sides  and 
4  feet  distant. 

53.  Find  the  volume  generated  by  an  ellipse  whose  axes  are 
40  inches  and  60  inches,  revolving  about  an  axis  in  its  own 
plane  whose  distance  from  the  center  of  the  ellipse  is  100 
inches. 

54.  AVhat  power  acting  horizontally  at  the  center  of  a  wheel 
4^  feet  in  diameter  and  weighing  270  pounds,  will  draw  it  over 
a  cylindrical  log  6  inches  in  diameter,  lying  on  a  horizontal 
plane  ? 

55.  Find  the  volume  generated  by  the  revolution  of  a  circle 
2  feet  in  diameter  about  a  tangent. 

56.  Find  the  surface  generated  by  the  revolution  of  a  circle 
2  feet  in  diameter  about  a  tangent. 

57.  Find  the  surface  and  volume  of  a  cylindric  ring,  the 
diameter  of  the  inner  circumference  being  12  inches  and  the 
diameter  of  the  cross  section  16  inches. 


54  MATHEMATICAL   WRINKLES 

58.  Eind  the  surface  and  volume  of  the  segment  of  the  same 
cylindric  ring,  if  a  plane  is  passed  perpendicular  to  its  axis, 
and  at  a  distance  of -4  inches  from  the  center. 

59.  A  galvanized  cistern  is  8  feet  in  diameter  at  the  top, 
10  feet  at  the  bottom,  and  10  feet  deep.  A  plane  passes  from 
the  top  on  one  side  to  the  bottom  on  the  other  side.  What  is 
the  volume  of  the  part  contained  between  this  plane  and  the 
base? 

60.  A   wineglass  in  the  form    of    a  frustum   of  a  cone  is 

4  inches  in  diameter  at  the  top,  2  inches  at  the  bottom,  and 

5  inches  deep.  If,  when  full  of  water,  it  is  tipped  just  so  that 
the  raised  edge  at  the  bottom  is  visible,  what  is  the  volume  of 
the  water  remaining  in  the  glass  ? 

61.  To  what  depth  will  a  sphere  of  cork,  2  feet  in  diameter, 
sink  in  water,  the  specific  gravity  of  cork  being  .25  ? 

62.  The  diameter  of  two  equal  circular  cylinders,  intersecting 
at  right  angles,  is  3  feet.    What  is  the  surface  common  to  both? 

63.  In  digging  a  well  4  feet  in  diameter,  I  come  to  a  log 
4  feet  in  diameter  lying  directly  across  the  entire  well.  What 
was  the  contents  of  the  part  of  the  log  removed  ? 

64.  What  is  the  volume  of  a  solid  formed  by  two  cylindric 
rings  2  inches  in  diameter,  whose  axes  intersect  at  right  angles 

and  whose  inner  diameters  are  10  inches  ? 

65.  Find  the  area  of  a  circular  lune  or  crescent 
ABCD;  the  chord  ^0=10  feet;  the  height 
EB  =  S  feet ;  and  the  height  ED  =2  feet. 

66.  Find  the  circumference  of  an  ellipse,  the 
transverse  and  conjugate  diameters  being  80 
inches  and  80  inches. 

67.  The  axes  of  an  ellipse  are  60  inches  and  20 
inches.  What  is  the  difference  in  area  between  the  ellipse  and 
a  circle  having  a  diameter  equal  to  the  conjugate  axis  ? 


MISCELLANEOUS  PROBLEMS  55 

68.  What  is  the  area  of  a  parabola  whose  base,  or  double 
ordinate,  is  30  inches  and  whose  altitude,  or  height,  is  20 
inches  ? 

69.  What  is  the  area  of  a  cycloid  generated  by  a  circle 
whose  radius  is  6  feet  ? 

70.  Two  men,  A  and  B,  started  from  the  same  point  at  the 
same  time ;  A  traveled  southeast  for  10  hours,  and  at  the  rate 
of  10  miles  per  hour,  and  B  traveled  due  south  for  the  same 
time,  going  6  miles  per  hour;  they  turned  and  traveled  directly 
towards  each  other  at  the  same  rates  respectively,  till  they 
met.     How  far  did  each  man  travel  ? 

71.  In  front  of  a  house  stand  two  pine  trees  of  unequal 
height;  from  the  bottom  of  the  second  to  the  top  of  the  first  a 
rope  80  feet  in  length  is  stretched,  and  from  the  bottom  of  the 
first  to  the  top  of  the  second  a  rope  100  feet  in  length  is 
stretched.  If  these  ropes  cross  10  feet  above  the  ground,  find 
the  distance  between  the  trees. 

72.  To  trisect  any  angle. 

73.  A  grocer  has  a  platform  balance  the  ratio  of  whose  arms 
is  9  to  10.  If  he  sells  20  pounds  of  merchandise  to  one  man, 
weighing  it  on  the  right-hand  pan,  and  20  pounds  to  another 
man,  weighing  it  on  the  left-hand  pan,  what  per  cent  does  he 
gain  or  lose  by  the  two  transactions  ? 

74.  A  and  B  carry  a  fish  weighing  54  pounds  hung  between 
them  from  the  middle  of  a  10-foot  oar.  One  end  of  the  oar 
rests  on  A's  shoulder,  but  the  other  end  is  pushed  1  foot  be- 
yond B's  shoulder.     What  part  of  the  weight  does  each  carry  ? 

75.  A  half-ounce  bullet  is  fired  with  a  velocity  of  1400  feet 
per  second  from  a  gun  weighing  7  pounds.  Find  the  velocity 
in  feet  per  second  with  which  the  gun  begins  to  recoil,  and  the 
mean  force  in  pounds'  weight  that  must  be  exerted  to  bring  it 
to  rest  in  4  inches. 


56  MATHEMATICAL   WRINKLES 

76.  A  bullet  fired  with  a  velocity  of  1000  feet  per  second 
penetrates  a  block  of  wood  to  a  depth  of  12  inches.  If  it  were 
fired  through  a  plank  of  the  same  wood,  2  inches  thick,  what 
would  be  its  velocity  on  emergence,  assuming  the  resistance  of 
the  wood  to  the  bullet  to  be  constant  ? 

77.*  A  horse  is  tied  to  one  corner  of  a  rectangular  barn  30 
by  40  feet.  What  is  the  surface  over  which  the  horse  can 
range  if  the  rope  with  which  he  is  tied  is  80  feet  long  ? 

78.*  How  many  acres  are  there  in  a  circular  tract  of  land, 
containing  as  many  acres  as  there  are  boards  in  the  fence 
inclosing  it,  the  fence  being  5  boards  high,  the  boards  8  feet 
long,  and  bending  to  the  arc  of  a  circle  ? 

79.*  A  thread  passes  spirally  around  a  cylinder  10  feet  high 
and  1  foot  in  diameter.  How  far  will  a  mouse  travel  in  unwind- 
ing the  thread  if  the  distance  between  the  coils  is  1  foot  ? 

80.  A  string  is  wound  spirally  100  times  around  a  cone 
100  feet  in  diameter  at  the  base.  Through  what  distance  will 
a  duck  swim  in  unwinding  the  string,  keeping  it  taut  at  all 
times,  the  cone  standing  on  its  base  at  right  angles  to  the  sur- 
face of  the  water  ? 

81.*  After  making  a  circular  excavation  10  feet  deep  and 
6  feet  in  diameter,  it  was  found  necessary  to  move  the  center 
3  feet  to  one  side,  the  new  excavation  being  made  in  the  form 
of  a  right  cone  having  its  base  6  feet  in  diameter  and  its  apex 
in  the  surface  of  the  ground.  Required  the  total  amount  of 
earth  removed. 

82.*  A  20-foot  pole  stands  plump  against  a  perpendicular 
wall.  A  cat  starts  to  climb  the  pole,  but  for  each  foot  it 
ascends,  the  pole  slides  one  foot  from  the  wall ;  so  that  when 
the  top  of  the  pole  is  reached,  the  pole  is  on  the  ground  at 
right  angles  to  the  wall.  Required  the  distance  through  which 
the  cat  moved. 

*  These  problems  are  from  *'  Finkel's  Solution  Book." 


MISCELLANEOUS  PROBLEMS  57 

83.  A  tree  96  feet  high  was  broken  by  the  wind  in  such  a 
manner  that  the  top  struck  the  ground  36  feet  from  the  foot  of 
the  tree.  If  the  parts  remained  connected  at  the  place  of 
breaking,  forming  with  the  ground  a  right  triangle,  how  high 
was  the  stump  ? 

84.  The  distance  around  a  rectangular  field  is  140  rods,  and 
the  diagonal  is  50  rods.     Find  its  length,  breadth,  and  area. 

85.  The  area  of  a  rectangular  field  is  30  acres,  and  its  diag- 
onal is  100  rods.     FiQd  its  length  and  breadth. 

86.  Two  trees  of  equal  height  stand  upon  the  same  level 
plane,  60  feet  apart  and  perpendicular  to  the  plane.  One  of 
them  is  broken  off  close  to  the  ground  by  the  wind,  and  in  fall- 
ing it  lodges  against  the  other  tree,  its  top  striking  20  feet 
below  the  top  of  the  other.     Find  the  height  of  the  trees. 

87.  A  square  field  contains  10  acres.  From  a  point  in  one 
side,  10  rods  from  the  corner,  a  line  is  drawn  to  the  opposite 
side  cutting  off  6J  acres.     How  long  is  the  line  ? 

88.  Find  the  edge  of  the  largest  hollow  cube,  having  the 
shell  three  inches  in  thickness,  that  can  be  made  from  a  board 
42J  feet  long,  2  feet  wide,  and  3  inches  thick. 

89.  A  circular  farm  has  two  roads  crossing  it  at  right  angles 
40  rods  from  the  center,  the  roads  being  60  and  70  rods  re- 
spectively, within  the  limits  of  the  farm.  Find  the  area  of  the 
farm. 

90.  The  longest  straight  line  that  can  be  stretched  in  a  cir- 
cular track  is  200  feet  in  length.     Find  the  area  of  the  track. 

91.  From  the  two  acute  angles  of  a  right  triangle  lines  are 
drawn  to  the  middle  points  of  the  opposite  sides ;  their  respec- 
tive lengths  are  V73  and  V52  feet.  Find  the  sides  of  the 
triangle. 

92.  A  wheel  of  uniform  thickness,  4  feet  in  diameter,  stands 
in  the  mud  1  foot  deep.  What  fraction  of  the  wheel  is  out  of 
the  mud  ? 


MATHEMATICAL  RECREATIONS 

1.  Mary  is  24  years  old.  She  is  twice  as  old  as  Ann  was 
when  Mary  was  as  old  as  Ann  is  now.     How  old  is  Ann  ? 

2.  There  is  a  great  big  turkey  that  weighs  10  pounds  and 
a  half  of  its  weight  besides.     What  is  its  weight? 

3.  With  6  matches  form  4  equilateral  triangles,  the  side 
of  each  being  equal  to  the  length  of  a  match. 

4.  One  tumbler  is  half  full  of  wine,  another  is  half  full  of 
water.  From  the  first  tumbler  a  teaspoonful  of  wine  is  taken 
out  and  poured  into  the  tumbler  containing  the  water.  A 
teaspoonful  of  the  mixture  in  the  second  tumbler  is  then  trans- 
ferred to  the  first  tumbler.  As  the  result  of  this  double  trans- 
action is  the  quantity  of  wine  removed  from  the  first  tumbler 
greater  or  less  than  the  quantity  of  water  removed  from  the 
second  tumbler  ? 

5.  (i)  Take  any  number;  (ii)  reverse  the  digits;  (iii)  find 
the  difference  between  the  number  formed  in  (ii)  and  the 
given  number;  (iv)  multiply  this  difference  by  any  number 
you  please ;  (v)  cross  out  any  digit  except  a  naught ;  (vi)  give 
me  the  sum  of  the  remaining  digits,  and  I  will  give  you  the 
figure  struck  out. 

6.  (i)  Take  any  number;  (ii)  add  the  digits;  (iii)  sub- 
tract the  sum  of  the  digits  from  the  given  number ;  (iv)  cross 
out  any  digit  except  a  naught;  (v)  give  me  the  sum  of  the 
remaining  digits,  and  I  will  give  you  the  figure  struck  out. 

58 


MATHEMATICAL  RECREATIONS 


59 


7.  Given  a  plank  12  inches  square,  required  to  cover  a 
hole  in  a  floor  9  inches  by  16  inches,  cutting  the  plank  into 
only  two  pieces. 

8.  Place  four  9's  in  such  a  manner  that  they  will  exactly 
equal  100. 

9.  The  square  is  8  inches  by  8  inches.  By  forming  the 
latter  figure  out  of  the  four  parts  of  the  square  it  is  found  to  be 


_i 

II/__-__ 

r  w  iO        <?•! 

-1   -±.-^1 

% .-'   ■_ 

^^^ 

p------^  -- 

^--^ 

5  inches  by  13  inches  and  contains  65  square  inches, 
does  the  other  inch  come  from  ? 


Where 


10.  A  teamster  brought  5  pieces  of  chain  of  3  links  each  to 
a  blacksmith,  and  asked  the  cost  of  making  them  into  one  piece 
of  chain.  The  blacksmith  replied,  "I  charge  2  cents  to  cut 
a  link  and  2  cents  to  weld  a  link."  The  teamster  remarked 
that  as  it  would  require  4  cuts  and  4  welds  the  charge  would 
be  16  cents.  "No,  you  are  mistaken,"  said  the  blacksmith, 
"  I  figure  it  but  12  cents."     Who  was  right  ? 

11.  The  Hake  and  the  Hound 

A  hare  is  10  rods  before  a  hound,  and  the  hound  can  run 
10  rods  while  the  hare  runs  1  rod.  Prove  that  the  hound  will 
never  catch  the  hare. 

Proof.  —  When  the  hound  runs  10  rods  the  hare  has  gone 
1  rod.  When  the  hound  goes  the  1  rod  the  hare  has  run  ^-^ 
of  a  rod,  and  when  the  hound  has  run  the  ^  oi  q.  rod  the  hare 


60  MATHEMATICAL   WRINKLES 

has  run  y^^  of  a  rod,  and  so  on.  Therefore,  the  hare  is  always 
a  fraction  of  a  rod  ahead  of  the  hound,  and  hence  the  hound 
will  never  catch  the  hare. 

12.  To  prove  that  1  equals  2. 

Let  X  =  1. 

Then  x^  =  x. 

x^  —  1  =  X  —  1. 
Factoring,  (x  -\-  l)(x  —  1)  =  x  —  1. 

Dividing,  a; -f- 1  =  1, 

But  x  =  l.    Therefore  1=2. 

13.  A  Young  Lady  to  Her  Lover — 

I  ask  you,  sir,  to  plant  a  grove 

To  show  that  I'm  your  lady  love. 
This  grove  though  small  must  be  composed 

Of  twenty-five  trees  in  twelve  straight  rows. 
In  each  row  five  trees  you  must  place 

Or  you  shall  never  see  my  face. 

14.  In  going  from  A  to  B,  through  mistake  I  take  the  road 
going  via  (7,  which  is  nearer  A  than  B  and  is  12  miles  to  the 


left  of  the  road  I  should  have  traveled.  After  reaching  B  I 
find  that  I  have  traveled  35  miles.  Find  the  distances  from  A 
to  B,  A  to  C,  and  C  to  B,  each  being  an  integer. 

15.  A  room  is  30  feet  long,  12  feet  wide,  and  12  feet  high. 
On  the  middle  line  of  one  of  the  smaller  side  walls  and  1  foot 
from  the  ceiling  is  a  fly.  On  the  middle  line  of  the  opposite 
wall  and  1  foot  from  the  floor  is  a  spider.  The  fly  being 
paralyzed  by  fear  remains  still  until  the  spider  catches  it  by 
crawling  the  shortest  route.     How  far  did  the  spider  crawl  ? 


MATHEMATICAL  RECREATIONS  61 

16.  A  train  1  mile  long  starts  from  the  station  at  Glady. 
The  engine  leaves  the  station  and  the  conductor  waits  until  the 
caboose  comes,  when  he  jumps  on  the  caboose  and  walks  for- 
ward over  the  train.  When  the  engine  reaches  the  next  station, 
Oxley,  4  miles  distant  from  Glady,  the  conductor  steps  off 
the  engine.  How  far  does  the  conductor  ride  and  how  far  does 
he  walk  ? 

17.  Zeno's  Paradoxes  on  Motion 

(a)  Since  an  arrow  cannot  move  where  it  is  not,  and  since 
also  it  cannot  move  where  it  is  (in  the  space  it  exactly  fills),  it 
follows  that  it  cannot  move  at  all. 

(6)  The  idea  of  motion  is  inconceivable,  for  what  moves 
must  reach  the  middle  of  its  course  before  it  reaches  the  end. 
Hence  the  assumption  of  motion  presupposes  another  motion, 
and  that  in  turn  another,  and  so  ad  infinitum. 

18.  I  have  only  $2  when  approached  by  a  friend  whom 
I  owe  $2.  The  friend  asks  for  what  I  owe  him,  so  I  give 
him  the  $2  and  remark  that  it  is  all  my  money.  My  friend 
sympathizing  with  me  in  my  poverty,  hands  me  back  a  dollar 
and  says,  "  I  will  mark  your  account  paid."  What  per  cent  did 
I  gain  by  the  transaction  ? 

19.  What  Were  Our  Ages  When  Married? 

When  first  the  marriage  knot  was   tied   between   my  wife 

and  me. 
Her  age  did  mine  as  far  exceed,  as  three  plus  three  does  three ; 
But  when  three  years  and  half  three  years  we  man  and  wife 

had  been. 
Our  ages  were  in  ratio  then  as  twelve  is  to  thirteen. 


112  yd. 

20.   Find  the  value  of  the  above  lot  at  $  1  per  square  yard. 


62  MATHEMATICAL   WRINKLES 

21.  How  much  dirt  is  there  in  a  hole  the  dimensions  of 
which  are  an  inch  ? 

22.  Which  is  correct  to  say,  Five  and  six  are  twelve,  or  to 
say,  Five  and  six  is  twelve  ? 

23.  Three  men,  A,  B,  and  C,  wish  to  divide  $60  among 
themselves  so  as  to  receive  a  third,  fourth,  and  fifth,  respec- 
tively.    How  much  should  each  receive  ? 

24.  A,  B,  and  C  are  in  partnership.  They  own  17  sheep. 
They  wish  to  divide  them,  —  one  to  get  ^,  one  to  get  -|,  and 
the  other  to  get  ^.  How  can  this  be  done  without  killing  a 
sheep  ? 

25.  If  6  cats  eat  6  rats  in  6  minutes,  how  many  cats  will  it 
take  to  eat  100  rats  in  100  minutes  ? 

26.  A  man  who  owned  a  piece  of  land  in  the  form  of  a 
square,  decided  to  divide  it  among  his  wife  and  four  sons,  so 
as  to  give  his  wife  \  in  the  shape  of  a  square  in  one  corner 
and  to  give  the  remaining  |  to  his  sons.  He  divided  the  land 
so  that  each  son  received  the  same  amount  of  land  and  the  four 
pieces  were  similar.     How  did  he  divide  it  ? 

27.  A  philosopher  had  a  window  a  yard  square,  and  it  let  in 
too  much  light.  He  blocked  up  one  half  of  it,  and  still  had  a 
square  window  a  yard  high  and  a  yard  wide.    Show  how  he  did  it. 

28.  Why  does  it  take  no  more  pickets  to  build  a  fence 
down  a  hill  and  up  another  than  in  a  straight  line  from  top  to 
top,  no  matter  how  deep  the  gully  ? 

29.  A  room  with  eight  corners  had  a  cat  in  each  corner, 
seven  cats  before  each  cat,  and  a  cat  on  every  cat's  tail.  How 
many  cats  were  in  the  room  ? 

30.  (i)  Take  any  number  of  three  unequal  digits;  (ii)  re- 
verse the  order  of  the  digits;  (iii)  subtract  the  number  so 
formed  from  the  original  number ;  (iv)  give  me  the  last  digit 
of  the  difference,  and  I  will  give  you  the  difference. 


MATHEMATICAL   RECREATIONS  63 

31.  Select  any  two  numbers,  each  of  which  is  less  than  10. 
(i)  choose  either  of  them  and  multiply  it  by  5 ;  (ii)  add  7  to 
the  result;  (iii)  double  this  result;  (iv)  to  this  add  the  other 
number ;  (v)  give  me  the  result,  and  I  will  give  you  the  numbers 
originally  selected,  and  also  tell  you  which  one  you  multiplied 
by  5. 

32.  (i)  Take  any  number  of  three  unequal  digits,  in  which 
the  first  and  last  differ  by  not  less  than  2;  (ii)  form  a  new 
number  by  reversing  the  order  of  the  digits ;  (iii)  take  the  dif- 
ference between  these  two  numbers ;  (iv)  form  another  num- 
ber by  reversing  the  order  of  the  digits  in  this  difference; 
find  the  sum  of  the  results  in  (iii)  and  (iv).  The  sum  will  be 
1089. 

33.  Write  down  a  number  of  three  or  more  figures,  divide 
by  9,  and  name  the  remainder;  erase  one  figure  of  the  number, 
divide  by  9,  and  tell  me  the  remainder,  and  1  will  tell  you  what 
figure  you  erased. 

34.  Let  a  person  write  down  a  number  greater  than  1  and 
not  exceeding  10;  to  this  I  will  add  a  number  not  exceeding 
10,  alternately  with  him ;  and,  although  he  has  the  advantage 
in  putting  down  the  first  number,  I  will  reach  the  even  hundred 
first. 

35.  A  boy  bought  a  pair  of  boots  for  $  2  and  gave  a  S  10 
bill  in  payment.  The  merchant  had  a  friend  change  the  bill, 
and  gave  the  boy  his  change.  The  boy  left  the  city  with  the 
boots  and  the  $8.  The  friend  returned  the  bill,  saying  it  was 
a  counterfeit,  and  the  merchant  had  to  give  him  good  money 
for  it.     What  was  the  merchant's  loss  ? 

36.  A  man  having  a  fox,  a  goose,  and  a  peck  of  corn  is 
desirous  of  crossing  a  river.  He  can  take  but  one  at  a 
time.  The  fox  will  kill  the  goose  and  the  goose  will  eat  the 
corn  if  they  are  left  together.  How  can  he  get  them  safely 
across  ? 


64  MATHEMATICAL   WKINKLES 

37.  Suppose  a  hole  to  be  cut  through  the  earth,  and  a  ball 
dropped  into  this  hole,  what  would  be  the  behavior  of  the  ball* 
and  where  would  it  come  to  rest  and  how  ? 

38.  A  man  died  leaving  his  wife  and  four 
children  a  piece  of  land  as  shown  in  the  figure. 
The  wife  is  to  have  J  in  the  shape  of  a  tri- 
angle. The  children's  parts  are  to  be  similar, 
and  equal  in  size.  How  must  the  land  be 
divided  ? 

39.  With  what  four  weights  can  you  weigh  any  number  of 
pounds  from  1  to  40  ? 

40.  Can  you  plant  19  trees  in  9  rows  with  5  trees  to  the 
row  ? 

41.  Do  figures  ever  lie  ? 

42.  Can  you  multiply  feet  by  feet  and  get  square  feet  ? 

43.  A  hunter  walked  around  a  tree  to  kill  a  squirrel ;  the 
squirrel  kept  behind  the  tree  from  the  hunter.  Did  he  go 
around  the  squirrel  ? 

44.  A  Fallacy. 

Let  a;  be  a  quantity  which  satisfies  the  equation 

e^  =  —  1. 

Squaring  both  sides,  e'^'^  =  1. 

.-.  2x'  =  0. 

.\x  =  0. 

But  e'^  =  - 1  and  e«  =  1.     .'.-1  =  1. 

45.  I  have  $10,000.  If  I  spend  half  of  this  sum  to-day  and 
half  of  the  remainder  each  day  following,  in  how  many  days 
will  I  have  no  money  ? 

46.  In  the  diagram,  DEF  is  a  railroad  with  two  sidings,  DBA 
and  FCA,  connected  at  A.    The  portion  of  the  rails  at  A  which 


MATHEMATICAL   RECREATIONS 


65 


is  common  to  the  two  sidings  is  long  enough  to  permit  of  a 
single  car  like  P  or  Q,  running  in  or  out  of  it ;  but  it  is  too 


HZI 


short  to  contain  the  whole  of  an  engine  like  Ji.  Hence  if  an 
engine  runs  up  one  siding,  such  as  DBA,  it  must  come  back 
the  same  way. 

Car  No.  1  is  placed  at  B,  car  No.  2  is  placed  at  C,  and  an 
engine  is  placed  at  E. 

By  the  use  of  the  engine  interchange  the  cars,  without 
allowing  any  flying  shunts. 

12  3  4  47.  Given  twelve  coins  arranged  as  in  the 
figure.     Can  you  move  them  so  as  to  have 

^  ^    five  on   a   side  instead   of   four,   not   being 

allowed  to  introduce  other  coins  or  to  de- 

11  6  ,  . 

•  •    stroy  the  given  square? 

•  •      •      •        48.   ^  and  B  have  an  8-gallon  cask  of  wine 

and  wish  to  divide  it  into  two  equal  parts. 
The  only  measures  they  have  are  a  5-gallon  cask  and  a 
3-gallon  cask.  How  can  they  di- 
vide it? 


49.  I  bought  a  horse  for  $90, 
sold  it  for  $  100,  and  soon  rep\ir- 
chased  it  for  $80.  How  much  did 
I  make  by  trading  ? 

60.  Stick  six  pins  in  the  dots  so 
that  no  two  are  connected  by  a 
straight  line. 


I 


66  MATHEMATICAL   WRINKLES 

51.    Let  X  and  y  be  two  unequal  numbers,  and  let  z  be  their 
arithmetical  mean. 

Then,  x  -\-y  =  2z. 

•••  («  ■i-y)(^-y)  =  2  z{x  -  y). 
.'.  x^—  ?/  =  2xz  —  2yz. 


,-.  x?-2 

xz  = 

f-2yz. 

.*. 

ar^ 

-2xz^ 

z'  = 

f-2yz-\-z\ 

.'.(X-', 

zf  = 

(y-zf. 

.'.  X- 

■■  z  = 
.  x  = 

y-z. 

y- 

52.    To  prove  — 

1: 

=  1. 

First  solution : 

a^ 
-a^' 

.-.(- 

—  a 

=  a. 

.*. 

-1 

=  1. 

Second  solution ; 

(- 

-V 

=  1. 

• 

•.21og(. 

-1) 
-1 

=  log  1  =  0. 

But 

e«  = 

=  1.     .-. 

-1 

=  1. 

53.  With  the  seven  digits,  9,  8,  7,  6,  5,  4,  0,  express 
three  numbers  whose  sum  is  82,  each  digit  being  used  only 
once,  and  the  use  of  the  usual  notations  for  fractions  being 
allowed. 

54.  With  the  ten  digits,  9,  8,  7,  6,  5,  4,  3,  2,  1,  0,  express 
numbers  whose  sum  is  unity,  each  digit  being  used  only 
once. 

55.  With  the  nine  digits,  9,  8,  7,  6,  5,  4,  3,  2,  1,  express 
four  numbers  whose  sum  is  100,  each  digit  being  used  only 
once. 

56.  With  the  ten  digits,  9,  8,  7,  6,  5,  4,  3,  2,  1,  0,  express 
zero,  each  digit  being  used  only  once. 


MATHEMATICAL   RECREATIONS  67 

57.  With  the  ten  digits,  9,  8,  7,  6,  5,  4,  3,  2,  1,  0,  express 
three  numbers  whose  sum  is  2^,  each  digit  being  used  only 
once. 

58.  In  the  accompanying  diagram 
the  letters  stand  for  various  towns  and 
the  lines  indicate  the  only  possible 
paths  by  which  a  person  may  travel. 

Show  how  a  person  may  start  from 
any  town  and  go  to  every  other  town 
once,  and  only  once,  and  return  to 
the  initial  town. 

59.  Anoarisalever  of  what  class? 

60.  A  man  hires  a  livery  team  to  drive  from  ^  to  C  via  B 
and  return  for  $3.  At<  By  midway  between  A  and  C,  he 
takes  a  passenger  to  C  and  back  to  B.  What  should  he  charge 
the  passenger? 

61.  Put  down  the  figures  from  1  to  9,  leaving  out  the  8,  thus : 

12345679 
Select  any  one  of  the  figures,  multiply  it  by  9,  then  multiply 
the  whole  row  by  that  product.     Tell  me  what  your  answer  is, 
and  I  will  tell  you  what  number  you  selected. 

62.  Say  to  one  person  : 

"  Think  of  a  number  less  than  10 ;  double  it;  add  16;  divide 
by  2;  subtract  your  first  number,  and  your  answer  is  8." 

Say  to  another : 

"Think  of  a  number  less  than  10;  double  it;  add  9;  divide 
by  2 ;  subtract  your  first  number,  and  your  answer  is  4^." 

You  can  go  on  indefinitely,  giving  these  mental  exercises,  no 
two  alike,  to  each  one  in  a  large  audience,  and  announce  the 
answer  as  quickly  as  they  get  it  themselves.  The  secret  is 
this :  the  final  answer  is  always  half  the  number  you  tell  them 
to  add. 


68  MATHEMATICAL   WRINKLES 

63.  If  a  hen  and  a  half  laid  an  egg  and  a  half  in  a  day  and 
a  half,  how  many  eggs  would  7  hens  lay  at  the  same  rate  in  6 
days  ? 

64.  What  is  the  shortest  distance  that  a  fly  will  have  to  go, 
crawling  from  one  of  the  lower  corners  of  a  room  to  the  op- 
posite upper  corner,  the  room  being  20  feet  long,  15  feet 
wide,  and  10  feet  high? 

65.  If  a  man  charges  $2  for  sawing  a  cord  of  wood  3 
feet  long  into  3  pieces,  what  should  he  charge  for  sawing  a 
cord  of  wood  6  feet  long  into  pieces  the  same  length  ? 

66.  Three  boys  having  10,  30,  and  50  apples  visit  a  city  and 
sell  them  at  the  same  rate  and  receive  the  same  amount  for 
them.  How  much  do  they  receive  for  the  apples  and  at  what 
rate  do  they  sell  them  ? 

67.  When  a  boy  see-saws  on  the  long  end  of  a  plank  he  bal- 
ances against  16  bricks,  but  if  he  sits  on  the  shorter  arm  of  the 
plank  and  places  the  bricks  on  the  other  end  he  balances 
against  just  11.  Find  the  boy's  weight  if  a  brick  weighs  equal 
to  a  three-quarter  brick  and  three  quarters  of  a  pound. 

68.  A  switch  to  a  single-track  railroad  is  just  long  enough 
to  clear  a  train  of  19  cars  and  a  locomotive.  How  can  two 
trains  of  19  cars  and  a  locomotive  each,  going  in  opposite 
directions,  pass  each  other,  if  a  third  train  of  equal  length 
stands  on  the  switch,  without  dividing  a  train  ? 

69.  A  boy  was  sent  to  a  spring  with  a  5  and  a  3  quart  measure 
to  procure  exactly  4  quarts  of  water.     How  did  he  measure  it  ? 

70.  What  is  the  greatest  number  which  will  divide  27,  48, 
90,  and  174  and  leave  the  same  remainder  in  each  case  ? 

71.  There  is  in  the  floor  of  a  granary  a  hole  2  feet  in  width 
and  15  feet  in  length.  How  can  it  be  entirely  covered  with  a 
board  3  feet  wide  and  10  feet  long,  by  cutting  the  board  only 
once? 


MATHEMATICAL  RECREATIONS  69 

72.  What  part  of  J  square  yard  is  ^  yard  square  ? 

73.  Can  you  take  1  from  19  and  get  20  ? 

74.  If  an  egg  weighs  8  ounces  and  half  an  egg,  what'  does 
an  egg  and  a  half  weigh  ? 

76.  How  would  you  arrange  the  figures  8,  6,  and  1  so  that 
the  whole  number  formed  will  be  divisible  by  6  ? 

76.  What  three  figures  multiplied  by  4  will  make  precisely 
5? 

77.  Mr.  Jackson  owns  a  square  farm  the  area  of  which  is  20 
acres;  near  each  corner  stands  a  large  tree  which  is  upon  a 
neighbor's  land.  How  may  he  add  to  his  farm  so  as  to  have  a 
square  farm  containing  40  acres  and  still  not  own  the  land 
upon  which  the  trees  stand  ? 

78.  A  gentleman  rented  a  farm,  and  contracted  to  give  a 
landlord  J  of  the  produce;  but  prior  to  the  dividing  of  the 
corn,  the  tenant  used  45  bushels.  When  the  general  division 
was  made,  it  was  proposed  to  give  to  the  landlord  18  bushels  of 
the  heap,  in  lieu  of  his  share  of  the  45  bushels  which  the  tenant 
had  used,  and  then  to  begin  and  divide  the  remainder  as  though 
none  had  been  used.     Would  the  method  have  been  correct  ? 

79.  What  is  the  difference  between  half  a  dozen  dozen,  and 
six  dozen  dozen  ? 

80.  What  is  the  difference  between  twice  twenty-five  and 
twice  five  and  twenty  ? 

81.  1x2x3x4x5x6x7x8x9x0  =  ? 

82.  If  you  were  required  to  sell  apples  by  the  cubic  inch, 
how  would  you  find  the  exact  number  of  cubic  inches  in  a 
dozen  dozen  ? 

83.  A  man  who  has  only  two  rows  of  corn  hires  A  and  B  to 
hoe  them.  A  hoes  three  hills  on  B's  row  and  then  begins  on 
his  own  row.     B  finishes  his  row  and  hoes  six  hills  on  A's  row, 


70  MATHEMATICAL   WRINKLES 

when  they  find  the  work  is  finished.  Whifch  man  'hoes  the 
more  and  how  much  more,  the  rows  containing  the  same 
number  of  hills  ? 

84.  Two  ducks  before  a  duck,  two  ducks  behind  a  duck,  and 
a  duck  in  the  middle,  are  how  many  ducks  ? 

85.  Can  you  write  30  with  3  equal  figures  ? 

86.  Add  1  to  9  and  make  it  20. 

87.  Twenty-one  ears  of  corn  are  in  a  hollow  stump.  How 
long  will  it  take  a  squirrel  to  carry  them  all  out  if  he  carries 
out  3  ears  a  day  ? 

88.  In  the  bottom  of  a  well  45  feet  in  depth  there  was  a 
frog  who  commenced  traveling  toward  the  top.  In  his  journey 
he  ascended  3  feet  every  day,  but  fell  back  2  feet  every  night. 
In  how  many  days  did  he  get  out  of  the  well  ? 

89.  How  many  quarter-inch  blocks  will  it  take  to  fill  an 
inch  hole  ? 

90.  Cut  a  piece  of  cardboard  121  inches  long  by  2  inches 
wide  into  4  pieces  in  such  a  manner  as  to  form  a  perfect  square, 
without  waste. 

91.  A  man  and  his  wife,  each  weighing  150  pounds,  with 
two  sons,  each  weighing  75  pounds,  have  to  cross  a  river  in  a 
boat  which  is  capable  of  carrying  only  150  pounds'  weight. 
How  will  they  get  across  ? 

92.  Two  men  laid  a  wager  as  to  which  could  eat  the  more 
oysters;  one  ate  ninety-nine,  and  the  other  a  hundred  and 
won.     How  many  did  both  together  eat  ? 

93.  Thrice  naught  is  naught,  what  is  the  third  of  infinity  ? 

94.  If  \  of  20  is  4,  what  will  i  of  10  be  ? 

95.  If  the  third  of  6  be  3,  what  must  the  fourth  of  20  be  ? 

96.  Write  24  with  3  equal  figures,  neither  of  them  being  8. 


MATHEMATICAL  RECREATIONS  71 

97.  If  you  cut  30  yards  of  cloth  into  one-yard  pieces,  and 
cut  1  yard  every  day,  how  long  will  it  take  ? 

98.  What  number  is  that  when  multiplied  by  18,  27, 36,  45, 
54,  G3,  72,  81,  and  99  gives  a  product  in  which  the  first  and 
last  figures  are  the  same  as  those  in  the  multiplier,  and  when 
multiplied  by  9,  and  90,  gives  a  product  in  which  the  last 
figures  are  the  same  as  those  of  the  multiplier  ? 

99.  Three  market  women,  having  severally  10,  30,  and 
50  oranges,  sold  them  at  the  same  rate,  and  received  the  same 
amount  of  money.  What  were  the  rates  and  the  amounts  each 
received  ? 

100.  Suppose  a  steamer  in  rapid  motion  and  on  its  deck  a 
man  jumping.  Can  he  jump  farther  by  leaping  the  way  the 
boat  is  moving,  or  in  the  opposite  direction  ? 

101.  After  killing  a  certain  number  of  cattle,  it  was  found 
that  twenty  fore  feet  remained.     How  many  head  were  killed  ? 

102.  Can  you  write  27  with  two  equal  figures  ? 

103.  AVhen  is  a  number  divisible  by  9  ? 

104.  Find  the  figure  that  may  be  placed  anywhere  in,  or 
before,  or  after,  the  number  302,011,  and  make  it  divisible  by  9. 

105.  In  a  lot  where  there  are  some  horses  and  grooms,  can 
be  counted  82  feet  and  26  heads.  How  many  horses  and 
grooms  are  in  the  lot  ? 

106.  If  a  herring  and  a  half  cost  a  penny  and  a  half,  how 
much  will  11  herring  cost? 

107.  What  number  is  it  when  divided  by  2,  3,  4,  5,  or  6, 
there  is  a  remainder  of  1,  but  when  divided  by  7,  there  is  no 
remainder? 

108.  A  cord  passing  over  a  pulley  hung  to  a  pair  of  cotton 
scales,  suspended  from  a  beam,  has  a  150-pound  weight  fas- 
tened to  one  end  and  the  other  fastened  to  an  immovable  iron 


72 


MATHEMATICAL  WRINKLES 


stake.  How  much  will  the  scales  register?  How  much  more 
will  they  register  if  a  100-pound  weight  is  hung  to  a  loop  in 
the  cord  halfway  between  the  pulley  and  the  stake  ? 

109.  Why  can  a  fat  man  swim  more  easily  than  a  lean  one? 

110.  A  rifle  ball  thrown  against  a  board  standing  edgewise 

will  knock  it  down ;  the  same 
bullet  fired  at  the  board  will 
pass  through  it  without  disturb- 
ing its  position.     Why  is  this? 

111.  Can  you  mark  seven 
numbers  by  moving  on  a 
straight  line  from  one  number 
to  another,  as  in  the  figure, 
marking  the  number  you  move 
to?  Do  not  start  twice  from 
the  same  number. 

112.  The  sum  of  four  figures  in  value  will  be 
About  seven  thousand  nine  hundred  and  three ; 
But  when  they  are  halved,  you'll  find  very  fair, 
The  sum  will  be  nothing,  in  truth,  I  declare. 

113.  A  fisherman,  being  asked  the  depth  of  a  lake,  replied: 
"  This  pole  when  standing  on  the  bottom  reaches  one  foot  out 
of  the  water,  but  if  the  top  is  moved  through  an  arc  of  30°, 
it  becomes  level  with  the  surface  of  the  water."  How  deep 
is  the  lake  ? 

114.  What  is  the  shape  of  a  square  inch  ?   Of  an  inch  square  ? 

115.  What  integer  added  to  itself  is  greater  than  its  square  ? 

116.  What  number  added  to  itself  is  equal  to  its  square? 

117.  What  number  is  it  that  can  be  multiplied  by  1,  2,  3,  4, 
5,  or  6,  and  no  new  figures  appear  in  the  results? 

118.  3  +  3-3  +  3x3-3-3x0  =  ? 


MATHEMATICAL   RECREATIONS  73 

119.  Write  any  number  of  yards,  feet,  and  inches.  Reverse 
this  and  subtract  from  the  original.  Reverse  the  remainder 
and  add  to  the  remainder.  The  sum  will  in  every  case  be  12 
yards,  1  foot,  11  inches.  The  number  of  inches  first  written 
should  not  exceed  the  number  of  yards. 

120.  The  Numbers  37  and  73 

When  the  number  37  is  multiplied  by  each  of  the  figures  of 
arithmetical  progression,  3,  6,  9,  12,  15,  18,  21,  24,  27,  all  the 
products  which  result  from  it  are  composed  of  three  repeti- 
tions of  the  same  figure ;  and  the  sum  of  those  figures  is  equal 
to  that  by  which  you  multiplied  the  37. 

37  37  37  37  37 

3  6  9  12  15 


111 

222 

333 

444 

555 

37 

37 

37 

37 

18 

21 

24 

27 

666 

777 

888 

999 

If  the  number  73  be  multiplied  by  each  of  the  numbers  of 
arithmetical  progression,  3,  6,  9,  12,  15,  18,  21,  24,  27,  the  six 
products  which  result  from  this  multiplication  are  terminated 
by  one  of  the  nine  different  figures,  1,  2,  3,  4,  5,  6,  7,  8,  9. 
These  figures  will  be  found  in  the  reverse  order  to  that  of  the 
progression. 

121.  Arrange  the  figures  1,  2,  3,  4,  5,  6,  7,  8,  and  9  so  their 
sum  will  be  100. 

122.  Arrange  the  first  sixteen  digits  in  a  square  so  that 
they  may  count  34  in  every  straight  line. 

123.  Arrange  the  figures  1  to  9,  inclusive,  in  a  triangle  so 
as  to  count  20  in  every  straight  line. 

124.  Arrange  the  figures  1  to  9,  inclusive,  in  a  circle,  using 
one  in  the  center,  so  as  to  count  15  in  every  straight  line. 


74 


MATHEMATICAL   WEINKLES 


125.  Arrange  the  figures  1  to  19,  inclusive,  in  a  circle,  using 
one  in  the  center,  so  as  to  count  30  in  every  straight  line. 

126.  Arrange  the  figures  1  to  9,  inclusive,  in  a  triangle,  so 
as  to  count  17  in  every  straight  line. 

127.  Arrange  the  figures  1  to  9,  inclusive,  in  a  square  so  as 
to  count  15  in  every  straight  line. 

128. 


25 

6 

7 

24 

3 

4 

10 

17 

12 

22 

5 

15 

13 

11 

21 

8 

14 

9 

16 

18 

23 

20 

19 

2 

1 

A  Bordered  Magic  Square 
I 
129.    "  If  you  multiply  the  number  of  Jacob's  sons  by  the 

number  of  times  which  the  Israelites  compassed  Jericho,  and 
add  to  the  product  the  number  of  measures  of  barley  which 
Boaz  gave  Kuth,  divide  this  by  the  number  of  Haman's  sons, 
subtract  the  number  of  each  kind  of  clean  beasts  that  went 
into  the  ark,  multiply  by  the  number  of  men  that  went  to 
seek  Elijah  after  he  was  taken  to  heaven;  subtract  from  this 
Joseph's  age  at  the  time  he  stood  before  Pharaoh,  add  the 
number  of  stones  in  David's  bag  when  he  killed  Goliath; 
subtract  the  number  of  furlongs  that  Bethany  was  distant 
from  Jerusalem,  divide  by  the  number  of  anchors  cast  out 
when  Paul  was  shipwrecked,  subtract  the  number  of  persons 
saved  in  the  ark,  and  the  answer  will  be  the  number  of  pupils 
in  my  Sunday-school  class."  How  many  pupils  are  in  the 
class  ? 


MATHEMATICAL  RECREATIONS  76 

130.  Magic  Age  Table 


1 

2 

4 

8 

16 

32 

3 

3 

5 

9 

17 

33 

5 

6 

6 

10 

18 

34 

7 

7 

7 

11 

19 

35 

9 

10 

12 

12 

20 

36 

11 

11 

13 

13 

21 

37 

13 

14 

14 

14 

22 

38 

15 

15 

15 

15 

23 

39 

17 

18 

20 

24 

24 

40 

19 

19 

21 

25 

25 

41 

21 

22 

22 

26 

26 

42 

23 

23 

23 

27 

27 

43 

25 

26 

28 

28 

28 

44 

27 

27 

29 

29 

29 

45 

29 

30 

30 

30 

30 

46 

31 

31 

31 

31 

31 

47 

33 

34 

36 

40 

48 

48 

35 

35 

37 

41 

49 

49 

37 

38 

38 

42 

50 

50 

39 

39 

39 

43 

51 

51 

41 

42 

44 

44 

52 

52 

43 

43 

45 

45 

53 

53 

45 

46 

46 

46 

54 

54 

47 

47 

47 

47 

55 

55 

49 

50 

52 

56 

56 

56 

61 

51 

53 

57 

57 

57 

63 

54 

54 

58 

58 

58 

55 

55 

55 

59 

59 

59 

67 

58 

60 

60 

60 

60 

69 

59 

61 

61 

61 

61 

61 

62 

62 

62 

62 

62 

63 

as 

63 

63 

63 

63 

Key  to  Table.  —  Add  together  the  figures  at  the  top  of  each 
column  in  which  the  age  is  found,  and  the  sum  will  be  the  age 
sought.  Example :  Hand  the  table  to  a  lady  and  request  her 
to  tell  you  in  which  column  or  columns  her  age  is  found ;  if 
she  says  the  first,  fourth,  and  fifth,  you  can  say  it  is  25  by 
mentally  adding  together  the  first  figures  of  those  three  col- 
umns, and  so  on  for  any  age  up  to  63. 


76  MATHEMATICAL   WRINKLES 

131.  How  TO  Tell  a  Person's  Age 

Let  the  person  whose  age  is  to  be  discovered  do  the  figuring. 
Suppose,  for  example,  if  it  is  a  girl,  that  her  age  is  16,  and 
that  she  was  born  in  May.  Let  her  put  down  the  number  of 
the  month  in  which  she  was  born  and  proceed  as  follows : 

Number  of  month 5 

Multiply  by  2 10 

Add  5 15 

Multiply  by  50 750 

Then  add  her  age,  16 766 

Then  subtract  365,  leaving    ....     401 
Then  add  115 516 

She  then  announces  the  result,  516,  whereupon  she  may  be 
informed  that  her  age  is  16,  and  May,  or  the  fifth  month,  is  the 
month  of  her  birth.  The  two  figures  to  the  right  in  the  result 
will  always  indicate  the  age,  and  the  remaining  figure  or  figures 
the  month  in  which  her  birthday  comes. 

132.  A,  B,  and  C  were  a  mile  at  sea  when  a  rifle  was  fired 
on  shore.  A  heard  the  report,  B  saw  the  smoke,  and  C  saw 
the  bullet  strike  the  water  near  them.  Who  first  knew  of  the 
discharge  of  the  rifle  ? 

133.  A  Queer  Trick  of  Figures 

Put  down  the  number  of  your  living  brothers. 
Double  the  number. 
.  Add  3. 
Multiply  the  result  by  5. 
Add  the  number  of  your  living  sisters. 
Multiply  the  result  by  10. 
Add  the  number  of  dead  brothers  and  sisters. 
Subtract  150  from  the  result. 

The  right-hand  figure  will  be  the  number  of  deaths. 
The  middle  figure  will  be  the  number  of  living  sisters. 
The  left-hand  figure  will  be  the  number  of  living  brothers. 


MATHEMATICAL   RECREATIONS  77 

134.  Find  perfect  square  numbers,  each  containing  all  the 
10  digits,  under  the  following  conditions : 

(1)  The  least  square  possible. 

(2)  The  greatest  square  containing  no  repeated  digit. 

(3)  The  least  square  which,  when  reversed,  is  still  a  square. 

(4)  The  least  square  which  is  unaltered  by  reversal. 

135.  A  house  and  a  barn  are  20  rods  apart;  the  house  is 
10  rods  and  the  barn  6  rods  from  a  straight  brook.  What  is 
the  length  of  the  shortest  path  by  which  one  can  go  from  the 
house  to  the  brook  and  take  water  to  the  barn  ? 

136.  A  and  B  dig  a  ditch  for  $10;  A  can  dig  as  fast  as  B 
can  shovel  out  the  dirt,  and  B  can  dig  twice  as  fast  as  A  can 
shovel.    How  should  they  divide  the  $  10  ? 

137.  Three  Series  of  Remarkable  Numbers 

1x9  plus  1  =  10 

12x9  plus  2  =  110 

123x9  plus  3  =  1110 

1234x9  plus  4  =  11110 

12345x9  plus  5  =  111110 

123456x9  plus  6  =  1111110 

1234567  X  9  plus  7  =  11111110 

12346678  x  9  plus  8  =  111111110 

123456789  x  9  plus  9  =  1111111110 

1x9  plus  2  =  11 

12x9  plus  3  =  111 

123x9  plus  4  =  1111 

1234x9  plus  5  =  11111 

12345x9  plus  6  =  111111 

123456  X  9  plus  7  =  1111111 

1234567  X  9  plus  8  =  11111111 

12345678  X  9  plus  9  =  111111111 

123456789  x  9  plus  10=  1111111111 


T8  MATHEMATICAL  WRINKLES 

1x8  plus  1  =  9 

12  X  8  plus  2  =  98 

123  X  8  plus  3  =  987 

1234  X  8  plus  4  =  9876 

12345  X  8  plus  5  =  98765 

123456x8  plus  6  =  987654 

1234567  X  8  plus  7  =  9876543 

12345678  x  8  plus  8  =  98765432 

123456789  x  8  plus  9  =  987654321 

138.  At  10  A.M.  a  train  leaves  London  for  Edinburgh  run- 
ning at  50  miles  an  hour.  At  the  same  time  another  train 
leaves  Edinburgh  for  London,  traveling  at  40  miles  an  hour. 
Which  train  is  nearer  London  when  they  meet  ? 

139.  The  asterisks  in  the  incomplete  sum  printed  below 
indicate  missing  figures.     Find  all  the  missing  figures. 

1*32271 

52*4 

63**74 

88*47 

305417 

2*3547* 


4,107,303 

140.    Determine  the  missing  digits  in  the  following  sum  in 
multiplication : 

1*46 
*5 
6730 
107*8 


114,410 


141.  In  a  long  division  sum  the  dividend  is  529,565,  and  the 
successive  remainders  from  the  first  to  the  last  are  246,  222, 
and  542.     Find  the  divisor  and  the  quotient. 


MATHEMATICAL   RECREATIONS  79 

142.  The  sura  of  two  numbers  consisting  of  the  same  three 
digits  in  reverse  order  is  1170,  and  their  difference  is  divisible 
by  8.     Find  the  numbers. 

143.  A  girl  was  given  a  number  to  multiply  by  409,  but  she 
placed  the  first  figure  of  her  product  by  4  below  the  second 
figure  from  the  right  instead  of  below  the  third.  Her  answer 
was  wrong  by  328,320.     Find  the  multiplicand. 

144.  I  have  a  board  1|  inches  thick,  whose  surface 
contains  49f  square  feet.  Find  the  edge  of  a  cubical  box 
made  of  it. 

145.  Write  one  billion  by  the  Roman  notation. 

146.  Each  of  two  sons  inherit  30  %,  and  each  of  two  daugh- 
ters 20%,  of  a  parallelogrammatic  plantation,  containing  100 
acres,  and  having  an  open  ditch  on  its  long  diagonal.  The 
four  divisions  are  to  corner  somewhere  in  the  ditch,  and  each 
is  to  have  a  side  of  the  plantation  in  its  boundary.  Locate 
this  common  corner. 

147.  Why  is  the  difference  between  any  common  number 
of  three  digits  and  one  containing  the  same  digits  in  reversed 
order,  always  divisible  by  9,  11,  and  the  difference  of  the  ex- 
treme digits  ? 

148.  Required  with  six  9*s  to  express  the  number  100. 

149.  The  Lucky  Number 

Many  persons  have  what  they  consider  a  "  lucky  "  number. 
Show  such  a  person  the  row  of  figures  subjoined : 

1,  2,  3,  4,  5,  6,  7,  9 
(consisting  of  the  numerals  from  1  to  9  inclusive,  with  the  8 
only  omitted),  and  inquire  what  is  his  lucky  or  favorite  num- 
ber. He  names  any  number  he  pleases  from  1  to  9,  say  7. 
You  reply  that,  as  he  is  fond  of  sevens,  he  shall  have  plenty 
of  them,  and  accordingly  proceed  to  multiply  the  series  above 


80  MATHEMATICAL   WRINKLES 

given  by  such  a  number  that  the  resulting  product  consists  of 
sevens  only. 

Required  to  find  (for  each  number  that  may  be  selected)  the 
multiplier  which  will  produce  the  above  result. 

150.  Eather  and  son  are  aged  71  and  34  respectively.  At 
what  age  was  the  father  three  times  the  age  of  his  son  ?  and 
at  what  age  will  the  latter  have  reached  half  his  father's  age  ? 

151.  There  is  a  number  consisting  of  two  digits ;  the  num- 
ber itself  is  equal  to  five  times  the  sum  of  its  digits,  and  if 
9  be  added  to  the  number,  the  position  of  its  digits  is  re- 
versed.    What  is  the  number  ? 

152.  The  Expunged  Numerals 

Given  the  sum  following : 

111 
333 
555 

777 
999 

Required,  to  strike  out  nine  of  the  above  figures,  so  that  the 
total  of  the  remaining  figures  shall  be  1111. 

153.  A  Grayson  County  widower  married  a  Denton  County 
wddow ;  each  had  children.  Ten  years  later  a  domestic  tornado 
prevailed  in  the  back  yard  in  which  the  present  family  of  a 
dozen  children  were  involved.  Mother  to  father  :  "  Your  chil- 
dren and  my  children  are  picking  at  our  children."  If  the 
parents  now  have  each  nine  children  of  their  own,  how  many 
came  into  the  family  in  these  ten  years  ? 

154.  Some  of  the  numbers  differing  from  their  logarithms 
only  in  the  position  of  the  decimal  point. 

log  1.3712885742  =  .13712885742 

log  237.5812087593  =  2.375812087593 

log  3550.2601815865  =  3.5502601815865 


MATHEMATICAL  RECREATIONS  81 

155.  Consecutive  numbers  whose  squares  have  the  same 
digits : 

132  =  169        157-  =  24649        913^  =  833569 
1 42  =  196        158^  =  24964        914=-'  =  835396 

156.  To  arrange  the  ten  digits  additively  so  as  to  make  100. 

157.  Express  the  numbers  from  1  to  30  inclusive  by  using 
for  each  number  four  4's. 

158.  Invert  the  figures  of  any  three-place  number;  divide 
the  difference  between  the  original  number  and  the  inverted 
number  by  9;  and  you  may  read  the  quotient  forward  or 
backward. 

159.  Write  a  number  of  three  or  more  places,  divide  by  9, 
and  tell  me  the  remainder ;  erase  one  figure,  not  zero,  divide 
the  resulting  number  by  9,  tell  me  the  remainder,  and  I  will 
tell  you  the  figure  erased. 

160.  Can  a  fraction  whose  numerator  is  less  than  its  de- 
nominator be  equal  to  a  fraction  whose  numerator  is  greater 
than  its  denominator? 

161.  Show  why  8  must  be  a  factor  of  the  product  of  any 
two  consecutive  even  numbers. 

162.  A  and  B  take  a  job  of  digging  potatoes  for  $  5.  B  can 
pick  up  as  fast  as  A  digs,  but  if  B  digs  and  A  picks  them,  B 
must  begin  digging  ^  day  before  A  begins  picking,  in  order 
that  each  may  complete  his  work  at  the  same  time.  How 
shall  they  divide  the  money  ? 

163.  A  and  B  are  employed  to  dig  a  ditch  100  rods  long  for 
$  200.  A  is  to  get  $  1.75  per  rod  and  B  $2.25  per  rod.  How 
much  will  each  have  to  dig  so  as  to  be  entitled  to  an  equal 
share  of  the  money  ? 

164.  If  an  egg  balances  with  three  quarters  of  an  egg  and 
three  quarters  of  an  ounce,  find  the  weight  of  an  egg. 


82  MATHEMATICAL   WRINKLES 

165.  A  farmer  had  six  pieces  of  chain  of  5  links  each, 
which  he  wanted  made  into  an  endless  piece  of  30  links. 
If  it  costs  a  cent  to  cut  a  link  and  costs  a  cent  to  weld  it,  what 
did  it  cost  him  ? 

166.  A  vessel  of  water  full  to  the  brim  weighs  20  pounds. 
A  5-pound  live  fish  is  put  into  the  vessel.  Has  the  weight 
of  the  vessel  of  water  been  increased  or  diminished  ? 

167.  What  is  the  most  economical  form  of  a  tank  designed 
to  hold  1000  cubic  inches  ? 

168.  "Johnnie,  my  boy,"  said  a  successful  merchant  to  his 
little  son,  "  it  is  not  what  we  pay  for  things,  but  what  we  get 
for  them  that  makes  good  business.  I  gained  ten  per  cent  on 
that  fine  suit  of  clothes,  while  if  I  had  bought  it  ten  per  cent 
cheaper  and  sold  it  for  twenty  per  cent  profit,  it  would  have 
brought  a  quarter  of  a  dollar  less  money.  Now,  what  did  I 
get  for  that  suit  ?  " 

—  From  "  Our  Puzzle  Magazine." 

169.  While  discussing  practical  ways  and  means  with  his 
good  wife.  Farmer  Jones  said :  "  Now,  Maria,  if  we  should  sell 
off  seventy-five  chickens  as  I  propose,  our  stock  of  feed  would 
last  just  twenty  days  longer,  while  if  we  should  buy  a  hundred 
extra  fowl,  as  you  suggest,  we  would  run  out  of  chicken  feed 
fifteen  days  sooner."     How  many  chickens  had  they  ? 

—  From  "Our  Puzzle  Magazine." 

170.  Suppose  that  a  bird  weighing  1  ounce  flies  into  a  box 
with  only  one  small  opening,  and  without  resting  continues  to 
fly  round  and  round  in  the  box ;  does  it  increase  or  lessen  the 
weight  of  the  box  ? 

171.  John  can  weed  a  row  of  potatoes  while  James  digs 
three ;  but  James  can  weed  a  row  while  John  digs  a  row.  If 
they  get  $  10  for  their  work,  how  should  it  be  divided  between 
them? 


MATHEMATICAL   RECREATIONS  83 

172.  The  Watch  Trick 

The  following  is  a  well-knowu  way  of  indicating  on  a  watch 
dial  an  hour  selected  by  a  person.  The  hour  is  tapped  by  a 
pencil  beginning  at  VII  and  proceeding  backwards  round  the 
dial  to  VI,  V,  IV,  etc.,  and  the  person  who  selected  the  number 
counts  the  taps,  reckoning  from  the  hour  selected.  Thus,  if 
he  selected  VIII,  he  would  reckon  the  first  tap  as  the  9th; 
then  the  20th  tap  as  reckoned  by  him  will  be  on  the  hour 
chosen. 

It  is  obvious  that  the  first  seven  taps  are  immaterial,  but  the 
eighth  tap  must  be  on  XII. 

173.  What  is  a  third  and  a  half  of  a  third  of  10  ? 

174.  (i)  Write  down  a  number  thought  of;  (ii)  add  or 
subtract  any  number  you  wish ;  (iii)  multiply,  or  divide  by 
any  number  you  wish ;  (iv)  multiply  by  any  multiple  of  9 ; 
(v)  cross  out  any  digit  except  a  naught ;  (vi)  give  me  the  sum 
of  the  remaining  digits,  and  I  will  give  you  the  figure  struck 
out. 

175.  A  banker  going  home  to  dinner  saw  a  $  10  bill  on  the 
curbstone.  He  picked  it  up,  noted  the  number,  and  went  home 
to  dinner.  While  at  home  his  wife  said  that  the  butcher  had 
sent  a  bill  amounting  to  $10.  The  only  money  he  had  was 
the  bill  he  had  found,  which  he  gave  to  her,  and  she  paid  the 
butcher.  The  butcher  paid  it  to  a  farmer  for  a  calf,  the  far- 
mer paid  it  to  the  merchant,  who  in  turn  paid  it  to  a  washer- 
woman, and  she,  owing  the  bank  a  note  of  $10  went  to  the 
bank  and  paid  the  note.  The  banker  recognized  the  bill  as 
the  one  he  had  found,  and  which  to  that  time  had  paid  S  50 
worth  of  debt.  On  careful  examination  he  discovered  that  the 
bill  was  counterfeit.  Now  what  was  lost  in  the  transaction, 
and  by  whom  ? 

176.  What  is  the  difference  between  a  mile  square  and  a 
square  mile  ? 


84  MATHEMATICAL    WKINKLES 

177.  A  Multiplication  Trick 

Here  is  a  little  trick  in  multiplication  that  may  amuse  you. 
Ask  a  friend  to  write  down  the  numbers  12345G79,  omitting 
the  number  8.  Then  tell  him  to  select  any  one  figure  from 
the  list,  multiply  it  by  9,  and  with  the  answer  to  this  sum  mul- 
tiply the  whole  list  —  thus  assuming  that  he  selects  either  the 
figure  4  or  6. 

Select  4  X  9  =  36.  Select  6  x  9  =  54. 

12345679  12345679 

36  54 

74074074         ^  49382716 

37037037  *  61728395 

444444444  66666666Q 

You  see  the  answer  of  the  sum  is  composed  of  figures  similar 
to  the  one  selected. 

178.  Cook  was  within  10  miles  of  the  north,  pole  and  Peary 
was  also  within  10  miles  of  the  pole,  but  20  miles  from  Cook. 
What  direction  was  Peary  from  Cook  ?  Suppose  Peary  threw 
a  ball  at  Cook  and  hit  him.  In  what  direction  did  the  ball 
go? 

179.  A  man  has  12  pieces  of  chain  of  3  links  each.  He 
takes  them  to  a  blacksmith  to  unite  them  into  one  circular  or 
endless  chain.  If  it  costs  2  cents  to  cut  a  link  and  2  cents  to 
weld  a  link,  what  should  the  blacksmith  charge  for  the  job  ? 

180.  Take  2  pennies,  face  np wards  on  a  table  and  edges  in 
contact.  Suppose  that  one  is  fixed  and  that  the  other  rolls  on 
it  without  slipping,  making  one  complete  revolution  round  it 
and  returning  to  its  initial  position.  How  many  revolutions 
round  its  own  center  has  the  rolling  coin  made  ? 

181.  From  six  you  take  nine  ; 
And  from  nine  you  take  ten ; 
Then  from  forty  take  fifty, 
And  six  will  remain. 


MATHEMATICAL  RECREATIONS  85 

182.  A  room  is  30  feet  loug,  12  feet  wide,  and  12  feet  high. 
At  one  end  of  the  room,  3  feet  from  the  floor,  and  midway 
from  the  sides,  is  a  spider.  At  the  other  end,  9  feet  from  the 
floor,  and  midway  from  the  sides,  is  a  fly.  Determine  the 
shortest  path  the  spider  can  take  to  capture  the  fly  by  crawling. 

183.  A  Geometrical  Paradox 

A  stick  is  broken  at  random  into  3  pieces.  It  is  possible 
to  put  them  together  into  the  shape  of  a  triangle  provided  the 
length  of  the  longest  piece  is  less  than  the  sum  of  the  other 
2  pieces ;  that  is,  provided  the  length  of  the  longest  piece  is 
less  than  half  the  length  of  the  stick.  But  the  probability  that 
a  fragment  of  a  stick  shall  be  half  the  original  length  of  the 
stick  is  \.  Hence  the  probability  that  a  triangle  can  be  con- 
structed out  of  the  3  pieces  into  which  the  stick  is  broken  is.^. 

184.  A  Geometrical  Fallacy 

Proposition.  —  All  triangles  are  isosceles. 

Given,   any  triangle  ABC. 

To  prove  triangle  ABC  is  isosceles. 

Proof.  —  Draw  ME  perpendicular  to  AB  at  the  mid-point 
of  AB',  and  draw  CO,  the 
bisector  of  the  angle*  C,  in- 
tersecting the  line  ME  in  O. 

Draw  the  perpendiculars, 
OF  and  OX,  to  the  sides  AC 
and  BC,  respectively. 

Then  0N=  OF. 

.-.  CF=:  CN. 

Join  A  and  0 ;  also  join  0  and  B. 

Then  AO  =  BO. 

.'.  the  triangles  AOF  smd  OB^Vare  congruent. 

(Being  right  triangles  having  AO  =  BO  and  0F=:  ON.) 

.'.  AF=BN. 

.-.  AF+FC=  CN+NB,  or  AC  ==  BC 


86  MATHEMATICAL   WRINKLES 

185.  Three  men  robbed  a  gentleman  of  a  vase  containing 
24  ounces  of  balsam.  While  running  away  they  met  in  a 
forest  with  a  glass  seller,  of  whom  in  a  great  hurry  they  pur- 
chased three  vessels.  On  reaching  a  place  of  safety  they 
wished  to  divide  the  booty,  but  they  found  that  their  vessels 
contained  5,  11,  and  13  ounces  respectively.  How  could  they 
divide  the  balsam  into  equal  portions  ? 

186.  A  man  bets  — th  of  his  money  on  an  even  chance  (say 
tossing  heads  or  tails  with  a  coin) ;  he  repeats  this  again  and 

again,  each  time  betting  — th  of  all   the  money  then  in   his 

m 

possession.     If,  finally,  the  number  of  times  he  has  won  is 

equal  to  the  number  of  times  he  has  lost,  has  he  gained  or  lost 

by  the  transaction  ? 

187.  What  like  fractions  of  a  pound,  of  a  shilling,  and  of  a 
penny,  when  added  together,  make  exactly  a  pound  ? 

188.  Required  to  subtract  45  from  45  in  such  a  manner  that 
there  shall  be  a  remainder  of  45. 

189.  Any  prime  number,  which,  divided  by  4,  leaves  a  re- 
mainder 1  is  the  sum  of  two  perfect  squares. 

Below  is  given  a  list  of  all  prime  numbers  below  400  which, 
being  divided  by  4,  leave  a  remainder  of  1 : 

5  =  4  + 1  =  22  +  12  97  =  81  +  16  =  92  +  42 

13  =  9  +  4  =  32  +  22  101  =  100  +  1  =  10'  +  1'. 

17  =  16  +  1  =  42  +  12  109  =  100  +  9  =  102  +  32 

29  =  25  +  4  =  52  +  22  113  =  64  +  49  =  82  +  72 

37  =  36  -f- 1  =  62  + 12  137  =  121  + 16  =  II2  +  42 

41  =  25  +  16  =  52  +  42  149  =  100  +  49  =  IO2  +  72 

53  =  49  +  4  =  72  +  22  157  =  121  -f-  36  =  II2  +  62 

61  =  36  4-  25  =  62  +  52  173  =  169  +  4  =  I32  +  22 

73  =  64  -f  9  =  82  +  32  181  =  100  -f  81  =  IO2  +  9^ 

89=64  +  25  =  82-1-52  193  =  144  +  49  =  122  +  72 


MATHEMATICAL  RECREATIONS  87 

197  =  196  -h  1  =  14»  +  1*  313  =  169  + 144  =  13^  + 12^ 

229  =  225  -h  4  =  15*  4-  2«  317  =  196  4  121  =  14=^  4-  H* 

233  =  169  4-64  =  13^  +  82  337  =  256  +  81  =  16*  +  9* 

241  =  225  4- 16  =  15^  4-  4»  349  =  324  +  25  =  18^  +  5' 

257  =  256  4- 1  =  16'  4- 1*  353  =  289  -h  64  =  17^  4-  8' 

269  =  169  +  100  =  132  4- 102  373  =  324  +  49  =  18'  4-  7^ 

277  =  196  -f  81  =  142  4. 9=^  389  =  289  + 100  =  17'  4- 10« 

281  =  256  4-  25  =  16'  4. 5»  397  =  361  4-  36  =  19'  4-  6» 
293  =  289  4-4  =  172  +  2' 

190.  Any  number,  less  the  sum  of  its  digits,  is  divisible 
by  9. 

Proof.  Let  a  represent  the  units,  b  the  tens,  c  the  hundreds, 
d  the  thousands,  and  so  on. 

Then,     a  units         =  a  units  =         0  +  a  units 

h  tens  =      10  6  units  =     9  6+6  units 

c  hundreds  =    100  c  units  =    99  c  +  c  units 

d  thousands  =  1000  d  units  =  999  d-j-d  units 

The  number  =  999  d+99  c+9  6+a+6+c+d  units 
The  sum  of  the  digits  =a+6+c+d  units.     Subtracting,  we 
liave  a  remainder  of  999  (i  + 99  c+9  6. 

Since  999  d+99c+96isa  multiple  of  9,  it  is  divisible  by  9. 

191.  Two  persons  were  born  Jan.  1,  1830,  and  both  died 
Jan.  1,  1885 ;  yet  one  lived  10  days  longer  than  the  other. 
Explain  how  this  could  be  possible. 

192.  Two  men  are  20  miles  apart.  They  walk  in  the  same 
direction,  at  the  same  rate  of  speed,  for  the  same  length  of 
time ;  they  are  then  30  miles  apart.  Show  three  ways  in  which 
this  could  be  possible. 

193.  Two  men  start  from  the  same  place  at  the  same  time 
and  go  in  the  same  direction  for  the  same  length  of  time  at 
the  same  rate  of  speed.  When  they  have  gone  ^  the  journey 
they  find  they  are  about  8000  miles  apart,  yet  they  complete 
their  journeys  at  the  same  time.     How  is  this  possible  ? 


88  MATHEMATICAL    WRINKLES 

194.  Every  direction  is  soutli  except  up  and  down.  Where 
am  I? 

195.  A  boy  plants  a  grain  of  corn  5  inches  under  the  soil. 
The  first  night  it  sprouts  and  grows  ^  the  distance,  and  con- 
tinues to  grow  i-  the  remaining  distance  each  night  following. 
How  long  before  it  will  come  up  ? 

196.  Sterling  Jones,  a  heavy  boy,  weighs  20  pounds  plus  { 
of  his  own  weight,  plus  i  of  his  own  weight,  plus  j\  of  his 
own  weight  ...  to  infinity.     W^hat  is  his  weight  ? 

197.  Express  the  number  10  by  using  five  9's  in  4  different 
ways. 

198.  The  Paradox  of  Tristram  Shaxdy 

Tristram  Shandy  took  2  years  writing  the  history  of  the 
first  2  days  of  his  life,  and  lamented  that,  at  this  rate,  material 
would  accumulate  faster  than  he  could  deal  with  it,  so  that  he 
could  never  come  to  an  end,  however  long  he  lived.  But  had 
he  lived  long  enough,  and  not  wearied  of  his  task,  then,  even 
if  his  life  had  continued  as  eventfuUy  as  it  began,  no  part  of 
his  biography  would  remain  unwritten.  For  if  he  wrote  the 
events  of  the  first  day  in  the  first  year,  he  would  write  the 
events  of  the  nth  day  in  the  wth  year,  hence  in  time  the  events 
of  any  assigned  day  would  be  written,  and  therefore  no  part 
of  his  biography  would  remain  unwritten. 

—  From  Ball's  "  Mathematical  Recreations  and  Essays." 

199.  Swift's  Biological  Difficulty 

Great  fleas  have  little  fleas  upon  their  backs  to  bite  'em. 

And  little  fleas  have  lesser  fleas,  and  so  ad  infinitum. 

And  the  great  fleas  themselves,  in  turn,  have  greater  fleas  to 

go  on; 
While  these  have  greater  still,  and  greater  still,  and  so  on. 

—  De  Morgan. 


MATHEMATICAL  RECREATIONS  89 

200.  A  couple  of  dice  are  thrown.  The  thrower  is  invited 
to  double  the  points  of  one  of  the  dice  (whichever  he  pleases), 
add  5  to  the  result,  multiply  by  5,  and  add  the  points  of  the 
second  die.  He  states  the  total,  when  any  one  knowing  the 
secret  can  instantly  name  the  points  of  the  two  dice.  How  is 
it  done  ? 

201.  Three  dice  are  thrown.  The  thrower  is  asked  to  mul- 
tiply the  points  of  the  first  die  by  2,  add  5  to  the  result,  mul- 
tiply by  5,  add  the  points  of  the  second  die,  multiply  the  total 
by  10,  and  add  the  points  of  the  third  die.  He  states  the 
total.     Name  the  points  of  the  three  dice. 

202.  A  man  has  21  casks.  Seven  are  full  of  wine ;  7  half  full, 
and  7  empty.  How  can  he  divide  them,  without  transferring 
any  portion  of  the  liquid  from  cask  to  cask,  among  his  three 
sons,  —  Sam,  John,  and  James,  —  so  that  each  shall  have  an 
equal  quantity  of  wine  and  also  an  equal  number  of  casks  ? 

203.  Three  beautiful  ladies  have  for  husbands  three  men, 
who  are  as  jealous  as  they  are  young,  handsome,  and  gallant. 
The  party  are  traveling,  and  find  on  the  bank  of  a  river,  over 
which  they  have  to  pass,  a  small  boat  which  can  hold  no  more 
than  two  persons.  How  can  they  cross,  it  being  agreed  that 
no  woman  shall  be  left  in  the  society  of  a  man  unless  her  hus- 
band is  present  ? 

204.  A  certain  number  is  divisible  into  four  parts,  in  such 
manner  that  the  first  is  500  times,  the  second  400  times,  and 
the  third  40  times  as  much  as  the  last  and  smallest  part. 
What  is  the  number  and  what  are  the  several  parts? 

206.  What  is  the  smallest  number  which,  divided  by  2,  will 
give  a  remainder  of  1;  divided  by  3,  a  remainder  of  2;  di- 
vided by  4,  a  remainder  of  3;  divided  by  5,  a  remainder  of  4; 
divided  by  6,  a  remainder  of  5 ;  divided  by  7,  a  remainder  of 
6 ;  divided  by  8,  a  remainder  of  7 ;  divided  by  9,  a  remainder 
of  8 ;  and  divided  by  10,  a  remainder  of  9  ? 

-  V, 


90  MATHEMATICAL   WRINKLES 

206.  Given,  five  squares  of  paper  or  cardboard,  alike  in  size. 
Required,  so  to  cut  them  that  by  rearrangement  of  the  pieces 
you  can  form  one  large  square. 

207.  Given  a  board  3  feet  long  and  1  foot  wide.  Required 
to  cover  a  hole  2  feet  by  1  foot  6  inches,  by  not  cutting  the 
board  into  more  than  two  pieces. 

208.  Given  a  board  15  inches  long  and  3  inches  wide. 
How  is  it  possible  to  cut  it  so  that  the  pieces  when  rearranged 
shall  form  a  perfect  square  ? 

209.  Place  the  numbers  1  to  19  inclusive  on  the  sides  of  the 
six  equilateral  triangles  which  form  a  regular  hexagon,  so 
that  the  sum  on  every  side  will  be  the  same. 

210.  15  Christians  and  15  Turks,  being  at  sea  in  one  and  the 
same  ship  in  a  terrible  storm,  and  the  pilot  declaring  a  neces- 
sity of  casting  one  half  of  those  persons  into  the  sea,  that 
the  rest  might  be  saved;  they  all  agreed  that  the  persons  to 
be  cast  away  should  be  set  out  by  lot  after  this  manner,  viz., 
the  30  persons  should  be  placed  in  a  round  form  like  a  ring, 
and  then  beginning  to  count  at  one  of  the  passengers,  and  pro- 
ceeding circularly,  every  ninth  person  should  be  cast  into  the 
sea,  until  of  the  30  persons  there  remained  only  15.  The 
question  is,  how  those  30  persons  should  be  placed,  that  the 
lot  might  infallibly  fall  upon  the  15  Turks  and  not  upon  any 
of  the  15  Christians. 

211.  Some  Very  Old  Problems 

Heap,  its  seventh,  its  whole,  it  makes  19. 
—  From  Ahmes,  Collection  of  Problems,  made  in  Egypt  between 
3400  B.C.  and  1700  b.c. 

212.  The  numbers  from  1  to  80  admit  of  being  formed 
about  a  point  as  common  center  into  four  pentagons,  such  that 
each  side  of  the  first  pentagon  from  within  contains  two  num 


MATHEMATICAL   RECREATIONS  91 

bers,  each  side  of  the  second  pentagon  four  numbers,  each  of 
the  third  six  numbers,  and  each  side  of  the  fourth,  outermost 
pentagon  eight  numbers.  The  sum  of  the  numbers  of  each 
side  of  the  second  pentagon  is  122,  the  sum  of  those  of  each 
side  of  the  third  pentagon  is  248,  and  that  of  those  of 
eacli  side  of  the  fourth  pentagon  254.  Furthermore,  the  sum 
of  any  four  corner  numbers  lying  in  the  same  straight  line 
with  the  center,  is  also  the  same;  namely,  92. 


26  54 

81  49 

16 

10  80 

86  44 

76  9 

70  72 

60  16  32 

71  66 

65  25  65  27 

45  37  2 

61  24 

11  a^  14 


20  ^  n 

^  56  69  43 


63 


35  21  64  48 

69  73 

67  58 

6  62  23  79 

75  67 

77  19  22  63  18  ® 

41  38 

46  S3 

12         39         68         74         42         13 
61  28 


4         29         34         7         78         47         52         3 

—  From  "  Essays  and  Recreations  "  by  Schubert. 


92  MATHEMATICAL   WEINKLES 

213.  A  mule  and  a  donkey  were  walking  along,  laden  with 
corn.  The  mule  says  to  the  donkey,  "If  you  gave  me  one 
measure,  I  should  carry  twice  as  much  as  you.  If  I  gave  you 
one,  we  should  both  carry  equal  burdens."  Tell  me  their  bur- 
dens, O  most  learned  master  of  geometry. 

—  A  riddle  attributed  to  Euclid.     From  "  Palatine  Anthology," 

300  A.D. 

214.  What  part  of  the  day  has  disappeared  if  the  time  left 
is  twice  two  thirds  of  the  time  passed  away  ? 

—  "  Palatine  Anthology,"  300  a.d. 

215.  The  square  root  of  half  the  number  of  bees  in  a  swarm 
has  flown  out  upon  a  jessamine  bush,  |^  of  the  whole  swarm 
has  remained  behind ;  one  female  bee  flies  about  a  male  that 
is  buzzing  within  a  lotus  flower  into  which  he  was  allured  in 
the  night  by  its  sweet  odor,  but  is  now  imprisoned  in  it.  Tell 
me  the  number  of  bees. 

—  From  "  Lilavati,"  a  Chapter  in  Bhaskara's  great  work,  written 

in  1150  A.D. 

216.  Find  the  keyword  in  the  following  problem  in  "Letter 
Division." 

CPN)AOUIERT(PCAAU 

cpy 

PIUI 
PUCN 


RRIE 
RNAN 


REER 
RNAN 

RIRT 

RCUK 
EUT 

Note.  — For  other  problems  of  this  kind,  see  "  Div-A-Let,"  by  W.  H. 
Vail,  Newark,  N.  J. 


MATHEMATICAL  RECREATIONS 


93 


217.  Demochares  has  lived  a  fourth  of  his  life  as  a  boy;  a 
fifth  as  a  youth ;  a  third  as  a  man ;  and  has  spent  13  years  in 
his  dotage.     How  old  is  he  ? 

—  From  a  collection  of  questions  by  Metrodorus,  310  a.d. 

218.  Beautiful  maiden  with  beaming  eyes,  tell  me,  as  thou 
understandest  the  right  method  of  inversion,  which  is  tlie  num- 
ber which  multiplied  by  3,  then  increased  by  }  of  the  product, 
divided  by  7,  diminished  by  ^  of  the  quotient,  multiplied  by 
itself,  diminished  by  52,  the  square  root  extracted,  addition  of 
8,  and  division  by  10,  gives  the  number  2  ? 

—  From  "Lilavati." 

219.  Given  a  piece  of  cardboard  in  the 
form  of  a  Greek  or  equal-armed  cross,  as 
shown  in  the  figure.  Required,  by  two 
straight  cuts,  so  to  divide  it  that  the  pieces 
when  reunited  shall  form  a  square. 


220.   To  show  geometrically  that  1  =  0. 

First  Solution.  Take  a  square  that  is  8  units  on  a  side,  and 
cut  it  into  three  parts,  A,  B,  and  C,  as  shown  in  the  left-hand 
figure.     Fib  these  parts  together  as  in  the  right-hand  figure. 


Now  the  square  is  8  units  on  a  side,  and  therefore  contains 
64  small  squares,  while  the  rectangle  is  9  units  long  and  7 
units  wide,  and  therefore  contains  63  small  squares. 

Each  of  the  figures  are  made  up  of  -4,  B,  and  G. 


94 


MATHEMATICAL   WRINKLES 


In  the  square 
In  the  rectangle 


^  +  5  +  0  =  64. 

.-.64  =  63. 
(Things  equal  to  the  same  thing  are  equal  to  each  other.) 

.'.1  =  0. 
(By  subtracting  63  from  each  side  of  the  equation.) 

Second  Solution.     Take  a  square  that  is  8  units  on  a  side, 
and  cut  it  into  three  parts,  A,  B,  and  (7,  as  shown  in  the  right- 
hand  figure.      Eit  these 
parts  together  as  in  the 
left-hand  figure. 

Now  the  square  is  8 
units    on    a    side,    and 
therefore     contains     64 
small  squares,  while  the 
rectangle  is  13  units  long  and  5  units  wide,  and  therefore  con- 
tains Q>b  small  squares. 

Each  of  the  figures  are  made  up  of  A,  B,  and  O. 

In  the  rectangle  A -{- B  -{-  C  =  65. 
In  the  square       A -\- B  -\- C  =  64:. 
.-.  65  =  64. 
.-.  1  =  0. 


"" 

"1 

■" 

*" 

7 

y 

\^ 

y 

8 

y 

A 

^ 

,^ 

1 

l<< 

^ 

y 

n 

^ 

± 

-L 

L. 

13 


221.    To  prove  that  1 

Let 
Then 
and 


Note.  — If  a  =  1,  1 


=  200. 
a  =  6  =  10. 

. •.!  =  «» -1-61 
.-.  1  =  10^  +  102. 
.-.  1  =  200. 
if  a  =  2,  1  =  8  ;  if  a  =  3,  1  =  18  ;  etc. 


MATHEMATICAL   RECREATIONS 


95 


*  222.    To  prove  that  1  =  2000. 

Let  a  =  6  =  10. 

Then  a^  -  h^  =  0, 

and  a«  -  6«  =  0. 

(Things  equal  to  the  same  things  are  equal  to  each  other.) 

.-.  l  =  ci3-f-6^ 
(Dividing  by  a'  —  6^) 

.-.  1  =  W  +  10». 

.-.  1  =  2000. 

Note.  — If  a  =  1,  1  =  2;  if  a  =  2,  1  =  16;  if  a  =  3,  1  =  54;  etc.    Also 
many  other  problems  may  be  made  similar  to  problems  Nos.  221  and  222. 


223. 


Another  Geometrical  Fallacy 


To  prove  that  it  is  possible  to  let  fall  two  perpendiculars  to 
a  line  from  an  external  point. 

Take  two  intersecting  circles  with  centers  0  and  0'.     Let 
one  point  of  intersection  be 
P,  and  draw  the  diameters 
PJfandPxV. 

Draw  MN  cutting  the 
circumferences  at  A  and  B. 
Then  draw  PA  and  PB. 

Since  Z  PBM  is  inscribed 
in  a  semicircle,  it  is  a  right 
angle.  Also  since  /.PAN 
is  inscribed  in  a  semicircle,  it  is  a  right  angle. 

.'.PA  and  PB  are  both  ±  to  MN. 

224.  Given  three  or  more  integers,  as  30,  24,  and  16;  re- 
quired to  find  their  greatest  integral  divisor  that  will  leave 
the  same  remainder. 

•  The  exposing  of  fallacies  has  been  left  to  the  student.  They  should  be 
studied  in  every  High  School  and  College.. 


96  MATHEMATICAL   WRINKLES 


225.    To  Prove  that  You  are  as  Old  as  Methuselah 

Proof  : 

Let 

X  =  Methuselah's  age. 

Let 

y  =  your  age. 

Let 

s  =  the  sum. 

Then 

x-\-y  =  s. 

...  (x-\-y)(x-y)=s(x-y). 

.  • .  x^  —  y^  =  sx—  sy. 

.'.  x^—  sx  —  y^  —  sy. 

s-                       s^ 
4                       4 

■■■('-S"-(-iJ- 

s             s 

.-.  x  =  y. 

226.  How  many  shoes  would  it  take  for  the  people  of  a 
town  if  one  third  of  them  had  but  one  foot  and  one  half  the 
remainder  went  barefoot  ? 

227.  The  Spider  and  the  Four  Gnats 

On  a  suspended  piece  of  glass  10  inches  long,  4  inches  wide, 
and  4  inches  high  is  a  spider  and  four  gnats.  The  spider  is 
on  one  end  ^  inch  from  the  bottom  and  midway  between  the 
sides.  The  gnats  are  on  the  other  end.  Three  of  them  are 
\  inch,  I  inch,  and  1  inch,  respectively,  from  the  top  and  mid- 
way between  the  sides.  The  fourth  is  1|^  inches  from  the 
top  and  on  an  edge. 

Determine  the  shortest  path  possible,  by  way  of  the  six 
faces  of  the  piece  of  glass,  for  the  spider  to  catch  the  four 
gnats  and  return  to  the  place  from  which  he  started. 

228.  What  difference  would  there  be  in  the  weight  of  a  per- 
fectly air-tight  bird  cage,  depending  on  whether  the  bird  were 
sitting  on  the  perch  or  flying  about  ? 


MATHEMATICAL   RECREATIONS 


97 


A-  V 


229.   To  prove  that  part  of  a  line  equals  the  whole  line. 
Take  a  triangle  ABC^  and  draw 
CP  ±  to  AB. 

From     C    draw     CX,     making 
Z  Ar\  =  /-B. 

Then   A  ABC    and    ACX   are   ^. 
similar. 

.-.  A  ABC'.  A  ACX=BC':  CX\ 
Furthermore,  A  ABC'.  A  ACX^^AB.AX. 
.'.BC^.CT^AB.AX, 
WJ':AB  =  ~CT'.AX. 

W^AC^+A^-^AB'AP, 
~CX''  =  AC''-^AT-2AX'AP, 
2AB'AP_~AC'-^AX*-2AX'AP 


or 

But 
and 


AC*-^Aff 


or 


or 


AC' 


AB 


AB 
-\-AB-2AP 


AX 


AX 
+  AX-2AP. 


iB-'^^'-AX     ^^' 


AC'-AB'AX^AC^-ABAX 
AB  AX 

..\AB  =  AX. 
—  From  Wentworth  and  Smith's  "  Geometry/ 


230.   To  prove  that  part  of  an  angle  equals  the  whole  angle. 
Take  a  square  ABCD,  and  draw  MM'P,  the  ±  bisector  of 
CD.    Then  MM'P  is  also  the  ±  bisector  of  AB. 

From  B  draw  any  line  BX  equal  to  AB. 
-^X        Draw  DX  and  bisect  it  by  the  ±  NP.     Since 
'/       DX  intersects  CD,  Js  to  these  lines  cannot  be 
/        parallel,  and  must  meet  as  at  P. 
^  Draw  PA,  PD,  PC,  PX,  and  PB. 

Since  MP  is  the  ±  bisector  of  CD,  PD  =  PC 


p 


98  MATHEMATICAL   WRINKLES 

Similarly,  PA  =  PB,  and  PD  =  PX. 
..PX=PD=Pa 

But  BX=BC  by  construction,  and  PB  is  common  to  A 
PBX  and  P5(7. 

.'.A  PBX  is  congruent  to  A  PBC,  and  Z  X5P  =  Z  CBP. 
.'.  the  whole  Z  XBP  equals  the  part,  Z  (7J5P. 

—  From  Wentworth  and  Smith's  '^  Geometry." 

231.  The  Four-color  Map  Problem 

Not  more  than  four  colors  are  necessary  in  order  to  color  a 
map  of  a  country,  divided  into  districts,  in  such  a  way  that  no 
two  contiguous  districts  shall  be  of  the  same  color. 

Probably  the  following  argument,  though  not  a  formal  dem- 
onstration, will  satisfy  the  reader  that  the  result  is  true. 

Let  A,  B,  C  be  three  contiguous  districts,  and  let  X  be  any 
other  district  contiguous  with  all  of  them.  Then  X  must  lie 
either  wholly  outside  the  external  boundary  of  the  area  ABO 
or  wholly  inside  the  internal  boundary ;  that  is,  it  must  occupy 
a  position  either  like  X  or  like  X'.  In  either  case  every  re- 
maining occupied  area  in  the  figure  is  inclosed  by  the  boun- 
daries of  not  more  than  three  districts;  hence  there  is  no 
possible  way  of  drawing  another  area  Y 
which  shall  be  contiguous  with  A,  B,  C, 
and  X.  In  other  words,  it  is  possible  to 
draw  on  a  plane  four  areas  which  are  con- 
tiguous, but  it  is  not  possible  to  draw  five 
such  areas. 

If  A,  B,  C  are  not  contiguous,  each  with 
the  other,  or  if  X  is  not  contiguous  with  A 
B,  and  C,  it  is  not  necessary  to  color  them 
all  differently,  and  thus  the  most  unfavora- 
ble case  IS  that  already  treated.  Moreover,  any  of  the  above 
areas  may  diminish  to  a  point  and  finally  disappear  without 
affecting  the  argument. 

That  we  may  require  at  least  four  colors  is  obvious  from. 


MATHEMATICAL  RECREATIONS 


% 


the  above  diagram,  since  in  that  case  the  areas  Aj  B,  C,  and  X 
would  have  to  be  colored  differently. 

A  proof  of  the  proposition  involves  difficulties  of  a  high 
order,  which  as  yet  have  baffled  all  attempts  to  surmount 
them.  — From  Ball's  "  Mathematical  Recreations." 

232.  RoMEO  AND  Juliet 

On  a  checker  board  are  located  two  snails.  They  are  Romeo 
and  Juliet.  Juliet  is  on  her  balcony  waiting  the  arrival  of 
her  lover,  but  Romeo  has 
been  dining  and  forgets, 
for  the  life  of  him,  the 
number  of  her  house. 
The  squares  represent 
sixty-four  houses,  and  the 
amorous  swain  visits 
every  house  once  and  only 
once  before  reaching  his 
beloved. 

Now  make  him  do  this 
with  the  fewest  possible 
turnings.  The  snail  can 
move  up,  down,  and  across 
the  board  and  through  the  diagonals.     Mark  his  track. 

—  From  "  Canterbury  Puzzles." 

233.  Find  the  exact  dimensions  of  two  cubes  the  sum  of 
whose  volumes  will  be  exactly  17  cubic  inches.  Of  course  the 
cubes  may  be  of  different  sizes. 

234.  I  have  two  balls  whose  circumferences  are  respectively 
1  foot  and  2  feet.  Find  the  circumferences  of  two  other  balla 
different  in  size  whose  combined  volumes  will  exactly  equal 
the  combined  volumes  of  the  given  balls. 

235.  Can  the  number  11,111,111,111,111,111  be  divided  by 
any  other  integer  except  itself  and  unity  ? 


■^ 

Vik- 

100 


MATHEMATICAL   WKIKKLES 


236.  My  friend  owns  a 
house  containing  16  rooms  as 
indicated  in  the  diagram. 

While  visiting  him  one  day, 
he  said  to  me,  "  Can  you  enter 
at  the  door  A  and  pass  out  at 
the  door  B  and  enter  every  one 
of  the  16  rooms  once  and  only 
once  ?  "  Show  how  I  might 
have  done  this. 

237.  Given  a  plank  contain- 
ing 169  square  inches  as  shown  below. 


Show  how  a  hole 
13  inches  square  may  be  covered 
by  cutting  the  plank  into  three 
pieces. 

238.  Given  a  piece  of  cloth  in 
the  shape  of  an  equilateral  tri- 
angle. Required  to  cut 
it  into  four  pieces  that 
may  be  put  together  and 
form  a  perfect  square. 


239.  A  Shokt  Method  of  Multiplication 

^a^ampZe.  —Multiply  41,096  by  83. 

The  answer  is  found  to  be  3,410,968  by  inspection.  It  will 
be  observed  that  the  answer  is  found  by  placing  the  last  figure 
of  the  multiplier  before  the  number  and  the  first  after  it.  Also 
if  we  prefix  to  41,096  the  number  41,095,890,  repeated  any  num- 
ber of  times,  the  result  may  always  be  multiplied  by  83  in  this 
peculiar  manner. 

8  multiplied  by  86  =  688. 

Also  to  multiply  1,639,344,262,295,081,967,213,114,754,098,- 
360,655,737,704,918,032,787  by  71,  all  you  have  to  do  is  to  place 
another  1  at  the  beginning  and  another  7  at  the  end. 


MATHEMATICAL   ^T^CiSliXTIONS 


101 


♦  240.  The  SquaI^k  fAh%j^*)y'  '   "       '    , 

To  prove  that  the  diagonal  of  any  square  field  equals  the 
sum  of  any  two  sides. 


100  rd 


Fia.  1. 


FiQ.  2. 


Fig.  3. 


Given  the  square  field  ABCD  with  a  side  equal  to  100  rods. 
The  distance  from  Aio  C  along  two  sides  is  200  rods. 

Now  in  Fig.  1  the  distance  from  Ato  C  along  t;he  diagonal 
path  is  200  rods.  In  Fig.  2  the  steps  are  -smaller,  yet  the  di- 
agonal path  is  200  rods  long.  In  Fig.  3  the  steps  are  very 
small,  yet  the  distance  must  be  200  rods  and  would  yet  be  if 
we  needed  a  microscope  to  detect  the  steps.  In  this  way  we 
may  go  on  straightening  out  the  zigzag  path  until  we  ulti- 
mately reach  a  perfect  straight  line,  and  it  therefore  follows 
that  the  diagonal  of  a  square  equals  the  sum  of  any  two  sides. 
Can  you  expose  the  fallacy  ? 

241.  Given  a  rectangular  block  of  wood  8  inches  by  4 
inches  by  3J  inches.  Required  to  cut  it  into  similar  blocks 
2\  inches  by  IJ  inches  by  1\  inches  with  the  least  possible 
waste.     How  many  blocks  can  be  had  ? 

A  Time  Problem 

242.  A  man  Who  carries  a  watch  in  which  the  hour,  minute, 
and  second  hands  turn  upon  the  same  center  was  asked  the 
time  of  day.  He  replied,  "  The  three  hands  are  at  equal  dis- 
tances from  one  another  and  the  hour  hand  is  exactly  20- 
minute  spaces  ahead  of  the  minute  hand."  Can  you  tell  the 
time? 


•  See  footnote,  page  95. 


102  MATBEMATICAL   WRINKLES 

-'  '   Tnti:  'Tj^ze  Planter 

243.  Are  you  a  practical  tree  planter?  If  so,  you  are 
requested,  (a)  to  show  how  sixteen  trees  may  be  planted  in 
twelve  straight  rows,  with  four  trees  in  every  row,  (b)  to  show 
how  sixteen  trees  may  be  planted  in  fifteen  straight  rows,  with 
four  trees  in  every  row. 

244.  Five  persons  can  be  seated  in  six  different  ways  around 
a  table  in  such  a  manner  that  any  one  person  is  seated  only 
once  between  the  same  two  persons.  Show  the  manner  of 
seating. 

245.  Seven  persons  may  be  seated  in  fifteen  different  ways 
around  a  table  in  such  a  manner  that  any  one  person  is  seated 
only  once  between  the  same  two  persons.  Show  the  ways  in 
which  they  might  be  seated. 

246.  On  his  morning  stroll,  Mr.  Busybody  encountered  a 
laborer  digging  a  hole.  ^'  How  deep  is  that  hole  ?  "  he  asked. 
"  Guess,"  replied  the  workingman,  who  stood  in  the  hole. 
"  My  height  is  exactly  five  feet  and  ten  inches." 

"  How  much  deeper  are  you  going  ?  " 

"I  am  going  twice  as  deep,"  rejoined  the  laborer,  "and  then 
my  head  will  be  twice  as  far  below  ground  as  it  now  is  above 
ground." 

Mr.  Busybody  wants  to  know  how  deep  that  hole  will  be 
when  finished. 

247.  One  night  three  men.  A,  B,  and  C,  stole  a  bag  of  apples 
and  hid  them  in  a  barn  over  night,  intending  to  meet  in  the 
morning  to  divide  them  equally.  Some  time  before  morning 
A  went  to  the  barn,  divided  the  apples  into  three  equal  shares 
and  had  one  apple  too  many,  which  he  threw  away.  A  took 
one  share  and  put  the  others  back  into  the  bag.  Soon  after  B 
came  and  did  exactly  as  A  had  done.  Then  came  C,  who  re- 
peated what  A  and  B  had  done  before  him.  In  the  morning 
the  three  met,  saying  nothing  of  what  they  had  done  during 


MATHEMATICAL  KECREATIONS 


103 


the  night.  The  remaining  apples  were  divided  into  three  equal 
shares,  with  still  one  apple  too  many.  How  many  apples  were 
there  in  the  bag  at  the  beginning  ? 

248.   The  following  diagram  represents  a  section  of  a  rail- 
way track  with  a  siding.     Eight  cars  are  standing  on  the  main 


line  in  the  order  1,  2,  3,  4,  5,  6,  7,  8,  and  an  engine  is  standing 
on  the  side  track.  The  siding  will  hold  five  cars,  or  four  cars 
and  the  engine.  The  main  line  will  hold  only  the  eight  cars 
and  the  engine.  Also  when  all  the  cars  and  the  engine  are  on 
the  •  main  line,  only  the  one  occupying  the  place  of  8  can  be 
moved  on  the  siding.  With  8  at  the  extremity,  as  shown, 
there  is  just  room  to  pass  7  on  the  siding.  The  cars  can  be 
moved  without  the  aid  of  the  engine. 

You  are  required  to  reverse  the  order  of  the  cars  on  the 
main  line  so  that  they  will  be  numbered  8,  7,  6,  5,  4,  3,  2,  1 ; 
and  to  do  this  by  means  which  will  involve  as  few  transfer- 
ences of  the  engine,  or  a  car  to  or  from  the  siding  as  are  possible. 


249. 


The  Mysterious  Addition 


To  express  the  sum  of  five  numbers,  having  given  only  the 
first. 

Have  a  person  write  a  number,  say  55,369.  Subtract  two 
from  the  number,  and  place  it  before  the  remainder,  giving 
255,367,  which  is  the  sum  of  the  numbers  to  be  added.     Each 


104  MATHEMATICAL   WRINKLES 

number  is  to  contain  the  same  number  of  figures  as         kk  oqq 
the  first.  _  3g|4g^ 

After  the  first  number  is  expressed  have  the  per-  g-i  ^o* 
son  write  the  second,  say  38,465.  Then  write  the  03  461 
third  yourself,  using  such  figures  in  the  number,  rrn  koq 
that  if  added  to  the  figures  in  the  number  above  — j^- — 
will  make  nine.  Have  the  person  write  the  fourth  ^i^^,obi 
number.  Then  write  the  fifth  yourself  in  the  same  way  as 
the  third.     These  numbers  added  will  give  the  required  sum, 

250.  At  the  close  of  four  and  a  half  months'  hard  work,  the 
ladies  of  a  certain  Dorcas  Society  were  so  delighted  with  the 
completion  of  a  beautiful  silk  patchwork  quilt  for  the  dear 
curate  that  everybody  kissed  everybody  else,  except,  of  course, 
the  bashful  young  man  himself,  who  kissed  only  his  sisters, 
whom  he  had  called  for,  to  escort  home.  There  were  just  a 
gross  of  osculations  altogether.  How  much  longer  would  the 
ladies  have  taken  over  their  needlework  task  if  the  sisters  of 
the  curate  referred  to  had  played  lawn  tennis  instead  of  at- 
tending the  meetings?  Of*  course  we  must  assume  that  the 
ladies  attended  regularly,  and  I  am  sure  that  they  all  worked 
equally  well.     A  mutual  kiss  counts  two  osculations. 

—  From  "  Canterbury  Puzzles." 

251.  The  Arithmetical  Triangle 

This  name  has  been  given  to  a  contrivance  said  to  have 
originated  or  to  have  been  perfected  by  the  famous  Pascal. 
1 


2 

1 

3 

3 

1 

4 

6 

4 

1 

5 

10 

10 

5 

1 

6 

15 

20 

15 

6 

1 

7 

21 

35 

35 

21 

7   1 

8 

28 

56 

70 

56 

28   8 

etc. 

etc. 

MATHEMATICAL   RECREATIONS  105 

This  peculiar  series  of  numbers  is  thus  formed  :  Write  do^vn 
the  numbers  1,  2,  3,  etc.,  as  far  as  you  please,  in  a  vertical  row. 
On  the  right  hand  of  2  place  1,  add  them  together,  and  place 
3  under  the  1 ;  then  3  added  to  3  =  6,  which  place  under  the 
3 ;  4  and  6  are  10,  which  place  under  the  6,  and  so  on,  as  far 
as  you  wish.  This  is  the  second  vertical  row,  and  the  third  is 
formed  from  the  second  in  a  similar  way. 

This  triangle  has  the  property  of  informing  us,  without  the 
trouble  of  calculation,  how  many  combinations  can  be  made, 
taking  any  number  at  a  time,  out  of  a  larger  number. 

Suppose  the  question  were  that  just  given ;  how  many  selec- 
tions can  be  made  of  3  at  a  time,  out  of  8  ? 

On  the  horizontal  row  commencing  with  8,  look  for  the  third 
number ;  this  is  56,  which  is  the  answer. 

252.  Twelve  nests  are  in  a  circle.  In  each  nest  is  only  one 
egg.  Required  to  begin  at  any  nest,  always  going  in  the  same 
direction,  and  pick  up  an  egg,  pass  it  over  two  other  eggs,  and 
place  it  in  the  next  nest.  This  process  is  to  be  continued  until 
six  eggs  have  been  removed  and  then  six  of  the  nests  should 
contain  two  eggs  each,  and  the  other  six  should  be  empty. 
Show  how  this  can  be  done  by  making  the  fewest  possible 
revolutions  around  the  nests. 

253.  A  man  in  a  city  skyscraper,  in  a  time  of  fire,  made  his 
escape  by  descending  on  a  rope.  He  was  300  feet  above  the 
ground  and  had  a  rope  only  150  feet  long  and  1^  inches  in 
diameter.  Show  how  he  made  his  escape  without  jumping 
from  the  window  or  dropping  from  the  end  of  the  rope. 

254.  A  German  farmer  while  visiting  town  bought  a  cask  of 
wine  containing  100  pints  of  pure  wine.  After  reaching  home 
he  hid  the  cask  in  his  barn  thinking  no  one  would  find  it. 
While  away  from  home  his  neighbor  found  the  cask  and  drew 
out  30  pints.  Each  time  he  drew  out  a  pint  he  replaced  it  with 
a  pint  of  pure  water  before  drawing  the  next  pint.  How  much 
wine  was  stolen  ? 


106 


MATHEMATICAL   WRINKLES 


255.  While  out  fishing  on  a  lake  in  a  small  boat  I  found 
myself  without  oars.  I  was  two  miles  from  shore.  I  had 
nothing  to  use  to  row  the  boat.  Besides  this  there  was  no 
current  to  help  me,  for  the  water  was  perfectly  smooth.  I  had 
nothing  in  the  boat  but  a  heavy  trot-line  one  inch  in  diameter 
and  six  large  fish.  I  could  not  swim  and  had  no  way  of 
securing  assistance.  Was  it  possible  for  me  to  reach  the  shore 
under  such  circumstances  ?     If  so,  how  ? 

256.  C's  age  at  A's  birth  was  5i  times  B's  age  and  now  is 
equal  to  the  sum  of  xV's  age  and  B's  age.  If  A  were  3  years 
younger  or  B  4  years  older,  A's  age  would  be  |  of  B's  age. 
Find  the  ages  of  A,  B,  and  C.     (Solve  by  arithmetic.) 

257.  What  is  the  smallest  sum  of  money  in  pounds,  shil- 
lings, pence,  and  farthings  that  can  be  expressed  by  using  each 

of  the  nine  digits,  1,  2,  3,  4,  5,  6,  7, 8,  and  9, 
once  and  once  only  ? 

258.     A  Eeversible  Magic  Square 

The  digits  0, 1,  2,  6,  and  8,  when  turned 
upside  down,  can  be  read,  0,  1,  7,  9,  and 
8.  It  will  be  observed  that  this  square 
when  turned  upside  down  is  still  magic. 

259.  To  prove  that  part  of  an 
angle  equals  the  whole  angle. 

Take  a  right  triangle  ABO 
and  construct  upon  the  hypote- 
nuse BC  an  equilateral  triangle 
BCD,  as  shown. 

On  CD  lay  off  OP  equal  to  CA. 

Through  X,  the  mid-point  of 
AB,  draw  PX  to  meet  CB  pro- 
duced at  Q.     Draw  QA. 

Draw  the  ±  bisectors  of  QA 
and  QP,  as  YO  and  ZO.     These 


29 

IZ 

61 

Z2 

Zl 

62 

19 

2Z 

12 

21 

ZZ 

69 

6Z 

Z9 

22 

II 

MATHEMATICAL  RECREATIONS  107 

must  meet  at  some  point  O  because  they  are  ±  to  two  inter- 
secting lines. 

Draw  OQ,  OA,  OP,  and  OC. 

Since  O  is  on  the  ±  bisector  of  QA,  .'.  OQ  =  OA, 

Similarly  OQ=OP,  and  .'.  OA  =  OP. 

But  CA  =  CP,  by  construction,  and  CO  =  CO. 

.-.  A  AOC  is  congruent  to  A  POC,  and  Z  ^CO  =  Z  PCO. 

260.  Another  Triangle  Fallacy 

To  prove  that  the  sum  of  two  sides  of  a  triangle  is  equal 
to  the  third  side. 

Let  ABC  be  a  triangle.  -^ ^ ^ -,Z> 

Complete     the     parallelo- 
gram  and   divide  the   diag-  ^^ ^iv\"'""/  / 


onal  AC  into  n  equal  parts.  /  /____N<C. 

Through  the  points  of  divi-       /  /^'       ^\ 

sion  draw  n  —  1  lines  parallel     ^ ' -^ ^ 

to  AB.  Similarly,  draw  n  —  1 
lines  parallel  to  BC.  AB  will  be  divided  into  7i  equal  parts. 
Also  BC  will  be  divided  into  n  equal  parts.  The  parallelo- 
gram is  now  divided  into  n^  equal  and  similar  parallelograms. 

Note.  —  The  diagram  is  drawn  for  n  =  3. 

Taking  the  small  parallelograms  of  which  the  segments  of 
AC  are  diagonals,  we  have 

AB  +  BC=AM+  EF-\-  GH+  ME -{■  FG -\-  HC. 

A  similar  relation  is  true,  however  large  n  may  be.  Now  let  n 
increase  indefinitely.  Then  the  lines  AM,  ME,  EF,  etc.,  will 
get  smaller  and  smaller.  Finally  the  points  MFH  will  ap- 
proach indefinitely  near  the  line  AC,  and  ultimately  will  lie  on 
it.  When  this  is  the  case  the  sum  of  AM  and  ME  will  be 
equal  to  AE,  and  similarly  for  the  other  similar  pairs  of  lines. 

Hence,  AM-^ME^-EF+FO  +  On-\-HC:=^AE^EO-\-GC, 

01  AB-\-BC=Aa 


108  MATHEMATICAL   WRINKLES 

The  Fourth  Dimension 

Geometry  as  studied  in  the  schools  is  divided  into  two  parts, 
Plane  Geometry,  or  Geometry  of  Two  Dimensions,  and  Solid 
Geometry,  or  Geometry  of  Three  Dimensions.  These  divisions 
naturally  suggest  an  infinite  number  of  divisions.  Consider- 
ing space  as  an  aggregate  of  points,  the  line  is  a  one-dimen- 
sional space,  a  plane  is  a  two-dimensional  space,  and  a  solid  is 
a  three-dimensional  space.  To  fix  exactly  the  position  of  a 
point  on  a  line,  it  is  only  necessary  to  have  one  number  giving 
its  distance  from  some  fixed  point.  To  fix  exactly  the  position 
of  a  point  in  a  plane,  it  is  necessary  to  start  from  a  known 
point  and  measure  in  two  given  perpendicular  directions.  To 
fix  exactly  the  position  of  a  point  in  a  solid,  it  is  necessary  to 
start  from  a  known  point  and  measure  in  three  perpendicular 
directions. 

Thus  to  locate  a  man  traveling  north  from  a  given  place  it 
is  necessary  to  know  only  the  distance  traveled.  To  locate  a 
man  traveling  on  the  sea  it  is  necessary  to  have  two  measure- 
ments given  —  his  latitude  and  longitude.  To  locate  a  man 
traveling  in  the  air  it  is  necessary  to  have  three  measurements 
given  —  his  latitude,  longitude,  and  his  distance  above  or  below 
the  sea  level. 

The  question  now  arises :  Why  may  there  not  be  a  space 
of  four  dimensions  and  thus  a  geometry  of  four  dimensions  in 
which  the  exact  position  of  a  point  may  be  determined  by 
measuring  in  four  perpendicular  directions  ?  This  question  is 
one  which  we  cannot  escape.  Paul  may  have  had  the  fourth 
dimension  in  mind,  when,  speaking  of  spiritual  life,  he  said, 
"  That  Christ  may  dwell  in  your  hearts  by  faith,  that  ye  being 
rooted  and  grounded  in  love,  may  be  able  to  comprehend  with 
all  saints  what  is  the  breadth,  and  length,  and  depth,  and 
height "  (Eph.  3  :  17,  18)  ;  or  when  he  wrote,  "  I  knew  a  man 
whether  in  the  body,  or  out  of  the  body,  I  cannot  tell,  how  that 
he  was  caught  up  into  paradise  and  heard  unspeakable  words  " 


MATHEMATICAL  RECREATIONS  109 

(2  Cor.  12 :  2,  3).  What  did  John  mean  when  he  "  was  in  the 
spirit  viewing  the  Heavenly  Jerusalem  "  and  said,  "  The  city 
lieth  foursquare"  (Rev.  21:  16)?  Was  Christ's  transfigured 
body  a  four-dimensional  body?  Was  his  resurrected  body 
which  appeared  in  the  midst  of  a  closed  room  a  four-dimen- 
sional body  ?     Was  the  ascension  a  like  disappearance  ? 

Although  these  questions  cannot  be  answered  by  man,  we 
are  certain  that  the  term  fourth-dimensional  came  to  us  from 
a  firm  believer  in  spiritual  life.  We  can  neither  prove  nor 
deny  its  existence.  If  a  physical  fourth  dimension  exists,  a 
three-dimensional  body  would  never  know  it,  nor  would  we 
have  any  way  of  finding  out. 

If  we  connect  all  points  of  our  space,  a  three-dimensional 
space,  with  an  assumed  point  outside  of  it,  then  the  aggregate 
of  all  the  points  of  the  connecting  lines  constitutes  a  four- 
dimensional  space,  or  hyperspace.  As  a  moving  point  gener- 
ates a  line,  as  a  line  moving  outside  itself  generates  a  surface, 
as  a  surface  moving  outside  itself  generates  a  solid,  just  so  a 
solid  moving  outside  of  our  space  would  generate  a  hypersolid, 
or  portion  of  hyperspace.  Hyperspace  itself  may  be  conceived 
as  generated  by  our  entire  space  moving  in  a  direction  not  con- 
tained in  itself,  just  as  our  space  may  be  generated  by  the 
similar  motion  of  an  unlimited  plane. 

Has  hyperspace  a  real,  physical  existence?  If  so,  our  uni- 
verse must  have  a  small  thickness  in  the  fourth  dimension ; 
otherwise,  as  the  geometrical  plane  is  assumed  to  be  without 
thickness,  our  world,  too,  would  be  a  mere  abstraction  (as, 
indeed,  some  idealistic  philosophers  have  maintained),  that  is, 
nothing  but  a  shadow  cast  by  a  more  real  fourth-dimensional 
world. 

Of  what  use  is  the  conception  of  hyperspace?  It  is  of 
importance  to  the  mathematician.  The  notion  of  such  a 
geometry  as  a  logical  system  of  theorems  involved  in  a  set  of 
axioms  is  important  to  the  student.  It  gives  a  deeper  insight 
into  geometry.     The  conception  of  space  to  which  these  geo- 


110  MATHEMATICAL   WEINKLES 

metries  apply  is  of  great  assistance  in  the  application  of  geome- 
try to  the  other  mathematics.  Especially  is  it  of  importance 
because  of  the  parallelism  between  algebra  and  geometry.  It 
has  very  appropriately  been  called  the  playground  of  mathe- 
matics. It  is  not  only  of  importance  to  the  mathematician, 
but  is  also  of  much  importance  to  the  philosopher,  psycholo- 
gist, and  scientist  in  general.  It  is  a  question  of  interest  to 
every  person. 

The  geometry  of  two  dimensions  is  more  extensive  than  the 
geometry  of  one  dimension.  Also  the  geometry  of  three 
dimensions  is  more  extensive  than  the  geometry  of  two  di- 
mensions, yet  nearly  everything  in  the  solid  is  more  or  less 
analogous  to  something  in  the  plane.  Just  so  geometry  of  four 
dimensions  would  be  still  more  extensive  than  geometry  of 
three  dimensions,  yet  very  closely  related  to  it.  For  example, 
the  circle  studied  in  a  geometry  of  one  dimension  has  very  few 
properties,  while  studied  in  a  geometry  of  two  dimensions  has 
a  center,  radii,  chords,  tangents,  etc.,  and  studied  in  a  geometry 
of  three  dimensions  has  further  numerous  geometrical  relations 
with  the  sphere,  cone,  cylinder,  etc. 

Let  us  conceive  of  a  space  of  but  one  dimension.  A  being 
in  such  a  space  would  be  limited  to  a  straight  line,  which  he 
would  conceive  as  extending  infinitely  in  both  directions.  If 
you  were  a  point  and  lived  on  a  straight  line  you  would  be  a 
one-dimensional  man.  You  could  not  move  in  two-dimensional 
space,  but  could  think  about  it.  If  you  were  in  two-dimen- 
sional space  you  would  never  know  it.  You  could  move  back- 
ward and  forward  only.  You  could  not  look  up  or  down,  nor 
from  side  to  side.  You  could  see  only  the  back  of  the  man's 
head  in  front  of  you.  You  could  never  turn  around  and  talk 
to  a  man  behind  you.  If  you  encountered  another  being, 
neither  could  pass  the  other. 

Conceive  of  a  world  of  but  two  dimensions  inhabited  by 
two-dimensional  beings.  Such  a  world  would  lie  in  a  single 
plane,  having  length  and  breadth,  but  no  thickness.     The  in- 


MATHEMATICAL   RECREATIONS  111 

habitants  of  this  region  might  be  thought  of  as  the  shadows  of 
three-dimensional  beings.  By  a  miracle  one  of  these  beings 
becomes  endowed  with  a  knowledge  of  three  dimensions.  He 
could  then  do  marvelous  things  in  the  eyes  of  his  neighbors. 
He  could  disappear  and  reappear  at  will.  The  strongest  prison 
could  not  hold  him.  By  moving  out  of  the  plane  in  which  he 
lives  he  could  look  down  into  the  dwellings  and  even  into  the 
insides  of  his  neighbors.  He  would  then  be  a  god  in  the  pres- 
ence of  the  inhabitants  of  flatland,  or  shadowland. 

If  you  lived  on  a  surface,  you  would  be  a  two-dimensional 
man.  You  would  have  no  thickness.  You  could  slide  around 
like  quicksilver.  You  would  be  a  flat  man  and  could  not 
understand  how  a  third  dimension  could  possibly  exist.  You 
could  pass  your  neighbors.  You  would  be  living  in  a  three- 
dimensional  world  and  never  know  it.  You  could  pass  through 
a  three-dimensional  being  and  never  know  it.  You  could  pass 
through  a  brick  wall  and  never  see  it.  You  could  not  move  in 
three-dimensional  space,  but  could  think  of  it.  Only  a  square 
or  circle  would  be  necessary  to  imprison  you.  You  could  see 
all  around  you  but  could  not  look  down  or  up.  If  imprisoned, 
a  being  in  our  space  by  lifting  could  liberate  you  and,  to  your 
friends,  you  would  have  made  a  miraculous  escape.  If  you 
should  attempt  to  imprison  a  three-dimensional  criminal  in 
your  two-dimensional  jail,  he  would  escape  by  stepping  over 
the  walls  of  your  prison  and  you  would  never  realize  how  he 
eluded  you. 

Now,  if  there  be  a  four-dimensional  world,  our  three-dimen- 
sional space  must  lie  in  its  midst.  All  people  would  then  be 
three-dimensional  shadows  of  four-dimensional  beings.  We 
could  only  become  endowed  with  four-dimensional  knowledge 
or  become  four-dimensional  beings  b}'  supernatural  means.  We 
could  move  in  a  four-dimensional  being,  and  not  understand 
how  such  a  thing  is  possible.  If  there  be  such  a  thing  as  a 
four-dimensional  being,  it  would  perhaps  assist  us  in  under- 
standing the  following  scripture,  "  That  they  should  seek  the 


112  MATHEMATICAL   WRINKLES 

Lord,  if  haply  they  might  feel  after  him,  and  find  him,  though 
he  be  not  far  from  every  one  of  us :  for  in  him  we  live,  and 
move,  and  have  our  being"  (Acts  17 :  27,  28). 

If  you  were  a  four-dimensional  creature,  no  three-dimen- 
sional prison  would  hold  you,  and  we  should  never  know  how 
you  made  your  escape.  You  could  take  money  from  a  locked 
safe  without  opening  the-  door.  You  could  place  a  plum 
within  a  potato  without  breaking  the  peeling.  You  could  fill 
a  completely  inclosed  vessel.  You  could  turn  a  hollow  rubber 
ball  inside  out.  You  could  remove  the  contents  of  an  egg 
without  puncturing  the  shell,  or  drink  the  wine  from  a  bottle 
without  drawing  the  cork. 


EXAMINATION  QUESTIONS 

ARITHMETIC 

Teachers'  Examination  Questions.  —  Texas 

1.  Write  the  analysis  of  each  of  the  following : 

(a)  A  boy  has  75  cents,  with  which  he  can  buy  5  melons. 
Find  the  average  price  of  a  melon.  ' 

(6)  A  boy  has  75  cents,  with  which  he  buys  melons  at  the 
average  price  of  5  cents  each.  How  many  melons  does  he 
buy? 

2.  A  trader  bought  a  plantation  at  $  14  per  acre,  and  sold 
it  for  $  15,824,  gaining  $  2  per  acre.     Find  the  cost. 

3.  Find  the  product  of  the  smallest  prime  number  greater 
than  153,  and  the  greatest  composite  odd  number  less  than  230. 

4.  From  the  sum  of  29f  and  42|,  take  the  difference  of  20J 
and  10^. 

5.  The  product  of  two  factors  is  ^ ;  one  of  the  factors  is  |. 
Find  the  other. 

6.  What  per  cent  is  gained  by  buying  wheat  at  62J  cents 
per  bushel  and  selling  at  67^  cents  ? 

7.  In  a  proportion  the  inverse  ratio  of  the  first  term  to  the 
second  term  is  3^;  the  fourth  term  is  160.  Find  the  third 
term. 

8.  Give  solution  and  analysis :  Find  the  present  worth  and 
true  discount  of  a  note  for  $  135.75,  due  1  year  8  months 
15  days  hence,  money  being  worth  8  %. 

113 


114  MATHEMATICAL   WRINKLES 

9.  What  may  X  offer  for  a  house  which  pays  $  895  rent 
per  year  that  he  may  receive  8  %  interest  on  the  investment  ? 

10.    Reduce  to  lowest  terms :  .66|;  .125;  .371 

Teachers'  Examination  Questions.  —  Ohio 

1.  Define  aliquot  part,  mean  proportional,  maker  of  a  note, 
denominate  number. 

2.  (a)  Give  the  table  of  liquid  measure ;  of  dry  measure. 
(5)  How  many  cubic   inches    in  a  dry  quart?    in  a  liquid 

quart  ? 

3.  A  man  bought  a  lot  8  rods  square  at  the  rate  of 
$  1000  an  acre.  He  fenced  it  in  at  an  average  cost  of  .35  cents 
a  yard.  He  then  sold  the  lot  through  an  agent  for  $  750,  pay- 
ing 2i  cfo  commission.     Find  the  man's  profit. 

4.  (a)  What  is  meant  by  "  paying  by  check  "  ? 

(h)  Suppose  that  you  sell  to  Charles  Ray  a  horse  for 
$  250  and  agree  to  give  him  5  %  off  for  cash.  You  receive 
in  payment  his  check  for  the  amount  on  some  bank  of  which 
you  know.  Write  the  check,  supplying  the  necessary  details, 
but  using  a  fictitious  name. 

(c)  How  could  this  check  be  transferred  to  another  person 
so  that  the  money  could  be  drawn  only  on  his  order  ? 

5.  (a)  A  man  wishes  to  build  a  house  28  feet  by  32  feet. 
He  needs  four  sills,  each  6  inches  by  8  inches,  to  put  under  the 
walls.     How  much  will  they  cost  at  $  18  per  M  ? 

(Jb)  How  many  feet  of  siding  are  necessary  for  this  house, 
supposing  it  to  be  18  feet  high,  the  siding  being  5  inches  wide 
and  laid  4  inches  to  the  weather,  no  allowance  being  made  for 
gables,  doors,  or  windows  ? 

6.  (a)  A  certain  district  contains  taxable  property  valued 
at  $  150,000.    The  board  of  education  has  built  a  schoolhouse 


EXAMINATION  QUESTIONS  115 

costing   $1800.     What  will  the  schoolhouse  cost  a  taxpayer 
whose  property  is  valued  at  $  4800  ? 

(6)  Express  a  tax  rate  of  one  mill  as  a  rate  per  cent. 

7.  Write  a  rule  for  finding  (a)  the  area  of  a  circle  when 
the  radius  is  given;  (b)  the  surface  of  a  sphere  when  the 
radius  is  given ;  (c)  the  volume  of  a  pyramid. 

8.  A  father  gave  his  son  his  promissory  note  for  $  225,  due 
when  the  son  became  21  years  old.  The  rate  of  interest  was 
6%,  and  when  the  note  became  due,  the  principal  and  inter- 
est together  amounted  to  $303.75.  How  old  was  the  son  when 
the  note  was  given  ? 

State  Certificate.  —  Kentucky 

1.  Given  the  dividend,  quotient,  and  remainder,  how  may 
the  divisor  be  found  ?  If  10  apples  be  divided  equally  among 
five  boys,  which  of  the  terms  in  the  division  are  concrete  and 
which  abstract  ? 

2.  What  term  is  the  base  (a)  in  commission  ?  (b)  in  in- 
surance? (c)  in  profit  and  lass?  (d)  in  interest?  (e)  in 
discount  ? 

3.  At  6  o'clock  A.M.  the  thermometer  indicated  20°  above 
zero;  at  12  o'clock  M.,  5°  above  zero;  at  6  o'clock  p.m.,  7°  be- 
low zero.  Find  the  average  temperature  from  the  three  ob- 
servations.    Explain  the  process. 

4.  The  sum  of  two  numbers  is  147J,  and  their  difference 
83^.     What  are  the  numbers  ? 

5.  If  equal  sums  be  put  at  interest  for  1  year  12  days,  at 
5J  f/o  and  7  %  per  annum,  the  difference  in  interest  received 
on  the  two  principals  will  be  $  7.65.  Find  the  sum  invested 
in  each  case. 

6.  Wheat  is  worth  90  cents  per  bushel,  and  a  field  yields 
21  bushels  per  acre,  at  a  cost  of  $  16.75  per  acre  for  cultivation. 


116  MATHEMATICAL  WRINKLES 

If  the  cost  of  cultivation  be  increased  20%,  and  the  yield  be 
thereby  increased  30  %,  what  is  the  net  gain  per  acre? 

7.  The  longitude  of  Pensacola,  Fla.,  is  87°  15'  West.  Find 
the  difference  between  standard  time  and  local  (Meridian) 
time  in  that  city. 

8.  The  proceeds  of  a  3  months'  note  discounted  at  bank  at 
6  %  per  annum,  the  day  it  was  made,  were  $  400.  Find  the 
face  of  the  note. 

^9.  A  contractor  in  building  two  residences  finds  that  the 
number  of  mechanics  employed  on  the  first  is  to  the  num- 
ber employed  on  the  second  as  7:4,  the  weekly  wages  paid 
individuals  on  the  first  to  those  on  the  second  as  8 :  7,  and  the 
time  each  mechanic  was  employed  on  the  first  to  that  on  the 
second  as  5  :  12.  Find  the  relative  cost  of  labor  on  the  two 
buildings. 

10.  How  many  trees  planted  33  feet  apart  will  be  required 
to  cover  10  acres  in  the  shape  of  a  rectangle  20  rods  wide,  if 
no  allowance  is  made  for  space  beyond  the  outside  rows  ? 

State  Examination.  —  Michigan 

1.  (a)  9  is  a  factor  of  a  number  if  it  is  a  factor  of  the  sum 
of  its  digits,  and  not  otherwise.     Prove. 

(6)  At  what  time  between  2  and  3  o'clock  are  the  minute 
and  hour  hands  at  right  angles  to  each  other  ? 

2.  In  a  circle  1  mile  in  diameter  three  circles  are  inscribed, 
tangent  to  one  another  and  touching  the  larger  circumference. 
What  is  the  area  of  the  space  inclosed  by  the  three  circles  ? 

3.  Which  would  be  the  better  investment  and  how  much 
better  for  a  capital  of  $5000:  Baltimore  &  Ohio  Eailroad 
stock  quoted  at  127|,  brokerage  ^  %,  paying  semiannual  divi- 
dends of  3^  %  and  the  balance  in  a  savings  bank  paying  3  %, 
or  the  whole  in  a  6  %  mortgage  ? 


EXAMINATION  QUESTIONS  117 

4.  Write  a  concrete  problem  involving  cube  root  and  solve 
in  full  as  you  would  require  your  pupils  to  solve. 

5.  Discuss  briefly  as  to  the  advisability  of  teaching  in  the 
grades :  metric  system,  compound  proportion,  equations,  cube 
root,  geometrical  constructions,  partnership,  longitude  and  time. 

6.  A  train  weighing  126  tons  rests  on  an  incline  and  is 
kept  from  moving  down  by  a  force  1500  pounds.  What  is  the 
grade? 

7.  Change  4321  from  scale  of  10  to  scale  of  8  and  explain. 

8.  Find  the  ratio  of  the  side  of  a  cube  to  the  radius  of  a 
sphere  if  the  volume  of  the  cube  is  twice  that  of  the  sphere. 

9.  Discuss  and  illustrate  graphic  arithmetic. 

10.  The  marbles  in  a  box  can  be  divided  exactly  into  groups 
of  17,  but  when  divided  into  groups  of  16,  18,  or  24,  9  remain 
in  each  case.     How  many  marbles  are  there  ? 

County  Examination.  —  Texas 

1.  Two  thirds  of  A's  money  equal  |  of  B's.  If  they  put 
their  money  together,  what  part  of  the  whole  will  A  own  ? 

2.  S  600.00  Dallas,  Texas,  Jan.  15,  1904. 
For  value  received  I  promise  to  pay  David  Dooley,  or  order, 

on  demand,   six   hundred   dollars,  with   interest  at  8  %  per 
annum. 

What  amount  will  pay  the  above  note  Aug.  20, 1904,  at  exact 
interest  ? 

3.  If  you  double  the  rate  and  time,  what  must  be  done  to 
the  principal,  that  the  interest  be  unchanged?  How  many 
terms  are  involved  in  interest  ?  At  what  rate  must  any  prin- 
cipal be  placed  to  make  5  times  itself  in  3  years  ? 

4.  A  is  in  40°  W.  longitude.  When  it  is  3  a.m.  at  A, 
where  must  5  be  in  order  that  it  may  be  10  i'.m.  ? 


118  MATHEMATICAL   WRINKLES 

5.  If  16  men  hoe  200  acres  of  cotton  in  15  days  of  8  hours 
each,  how  many  boys  can  hoe  150  acres  in  12  days  of  6  hours 
each;  provided,  that  while  working  a  boy  can  do  only  -J  as 
much  as  a  man,  and  that  the  boys  are  idle  ^  of  the  time  ? 

6.  A  miller  charges  ^  toll  for  grinding  corn.  How  many 
bushels,  pecks,  and  quarts  must  a  man  take  to  mill  in  order 
that  he  may  obtain  13  bushels  of  meal  ? 

7.  The  solid  contents  of  a  cube  and  of  a  sphere  are  each 
3,048,625  cubic  inches.  Which  has  the  greater  surface,  and 
how  much  greater  ? 

8.  The  ice  on  a  circular  lake  is  1^  feet  thick.  If  the  lake 
is  1000  yards  in  circumference,  how  many  cubic  feet  of  ice  on 
the  lake  ? 

9.  I  bought  two  houses  for  $  1800,  paying  25  %  more  for 
one  than  for  the  other.  I  sold  the  cheaper  house  at  a  profit 
of  20  %,  and  the  higher  priced  house  at  a  loss  of  16|  %.  How 
many  dollars  did  I  gain  or  lose  ?  What  was  my  gain  or  loss 
per  cent  ? 

10.  A  bookseller  buys  a  book  whose  catalogue  price  is  $  4 
at  a  discount  of  25  %,  20  %,  and  8^  %,  and  sells  it  at  10  % 
above  the  catalogue  price.  What  per  cent  profit  does  he 
make  ? 

Commercial  Arithmetic.  —  Indiana 

1.  Illustrate  checking  residts  by  9's  and  ll's. 

2.  A  farmer  wishes  to  construct  a  square  granary  18  feet  on 
each  side  that  will  hold  800  stricken  bushels.  Find  the  depth 
of  the  bin  by  the  approximate  rule. 

3.  Illustrate  a  calculation  table. 

4.  A  man  had  6  acres  of  land ;  to  one  party  he  sold  a  piece 
25  rods  by  20  rods,  and  to  another  party  140  square  rods. 
What  per  cent  of  the  field  remained  unsold  ? 


EXAMINATION   QUESTIONS  119 

5.  Define  the  following:  Discount  series^  gross  pricey  net 
price, 

6.  Make  a  copy  of  a  bill  of  goods  showing  the  purchase  of 
four  articles,  one  article  at  a  discount  of  5  %  ;  the  second 
article,  10  %  ;  the  third  article,  15  % ;  the  fourth  article,  20  %. 

7.  Illustrate  a  cost  key  and  also  a  selling  key. 

8.  A  note  for  $  1600,  dated  Jan.  1,  1906,  bearing  interest  at 
6  %,  had  payments  indorsed  upon  it  as  follows :  March  1,  1906, 
$  250 ;  July  1,  1906,  $  25 ;  Sept.  1,  1906,  $  515  ;  Nov.  1,  1906, 
$  175.  How  much  was  due  upon  the  note  at  final  settlement, 
April  1,  1907  ? 

State  Certificate.  —  Ohio 

1.  The  sum  of  two  numbers  is  546,  their  G.  C.  D.  is  21,  and 
the  difference  of  the  other  two  factors  is  8.     Find  the  numbers. 

2.  At  what  two  times  between  4  and  5  o^clock  are  the  min- 
ute and  hour  hands  of  a  clock  equally  distant  from  4  ? 

3.  Certain  employees,  having  a  9-hour  day,  strike  because  of 
a  proposed  reduction  of  10  %  in  wages.  They  resume  work  at 
the  same  wages,  but  have  a  longer  day.  If  the  increase  in 
time  is  (to  the  firm)  equivalent  to  the  proposed  cut  of  10  %,  by 
what  per  cent  are  the  hours  increased  ? 

4.  A  dealer  sells  an  article  at  a  gain  of  10  %  ;  had  he  paid 
for  it  16  I  %  less,  and  sold  it  for  7  cents  less,  he  would  have 
gained  25%.     End  the  cost. 

5.  A  man  agrees  to  pay  S  6000  for  a  lot  in  three  equal  pay- 
ments, including  6  %  interest  on  unpaid  money.  What  is  the 
yearly  payment  ? 

6.  A  lady  buys  20  yards  of  cloth  for  $  20;  for  some  she  pays 
\  oi  a.  dollar  a  yard,  for  some  |  of  a  dollar  a  yard,  and  for  the 
remainder  $4  a  yard.  How  many  yards  of  each  kind  did  she 
buy,  provided  she  bought  a  whole  number  of  yards  of  each  ? 


120  MATHEMATICAL   WRINKLES 

7.  A  board  is  6  inches  wide  at  one  end  and  18  inches  wide 
at  the  other  end.  If  it  is  16  feet  long,  how  far  from  the  shorter 
end  must  it  be  cut,  parallel  to  the  ends,  to  divide  it  into  two 
equal  parts? 

8.  A  man  has  a  square  tract  of  land  which  contains  as  many 
acres  as  it  requires  rails  to  build  a  fence  around  it.  If  the 
fence  is  four  rails  high,  and  the  rails  are  12  feet  long,  how 
many  acres  are  in  the  field  ? 

9.  Pure  ground  mustard  contains  35  %  of  oil.  A  sample 
of  mustard  is  adulterated  with  wheat  flour.  The  per  cent 
of  oil  found  in  a  sample  is  15.  Find  the  per  cent  of  wheat 
flour  in  the  mixture,  allowing  2  %  of  oil  to  exist  naturally  in 
wheat  flour. 

10.   The  true  discount  of  a  certain  sum  for  one  year  is  {^ 
of  the  interest.     Find  the  rate. 

Teachers'  Examination. — Missouri 

1.  A  dealer  bought, two  horses  at  the  same  price.  He  sold 
one,  at  a  profit  of  20  %,  for  $102.  The  other  he  sold  at  a  loss 
of  10  %.     How  much  did  he  receive  for  the  latter? 

2.  A  rectangular  aquarium  is  32  inches  long,  24  inches 
wide,  and  16  inches  deep.  How  many  goldfish  may  be  kept  in 
it,  allowing  1  gallon  of  water  per  fish? 

3.  A  man  left  St.  Louis  and  traveled  until  his  watch  was  1 
hour  and  3  minutes  slow.  How  many  degrees  had  he  traveled 
and  in  what  direction  ? 

4.  The  base  of  a  triangular  field  is  360  yards,  and  the  altitude 
is  615  feet.     How  many  acres  does  it  contain  ? 

5.  Two  metal  spheres  of  the  same  material  weigh  1000  pounds 
and  64  pounds  respectively.  The  radius  of  the  second  is  1  foot. 
Find  the  radius  of  the  first. 


EXAMINATION  QUESTIONS  121 

6.  A  dealer  sold  an  automobile  for  $1000,  receiving  $400 
iu  cash  and  a  note  for  the  rest,  due  in  3  years,  interest  6%, 
payable  semiannually.  How  much  interest  was  paid  on  the 
note? 

7.  Which  is  the  better  investment  and  how  much,  5% 
bonds  at  110  or  6  %  bonds  at  118? 

8.  Name  some  subjects  given  in  arithmetic  that  you  think 
might  be  properly  omitted.     Give  reasons  for  your  answer. 

9.  What  must  be  invested  in  railroad  4^%  bonds  at  91|% 
to  yield  an  annual  income  of  $  1350,  brokerage  at  |^  %  ? 

10.  Analyze:  ^  of  the  price  paid  for  a  cow  was  f  of  the  cost 
of  a  horse.  The  horse  cost  $99  more  than  the  cow.  •  Find  the 
cost  of  each. 

State  Examination. — New  York 

1.  What  rate  per  cent  of  profit  will  a  man  make  by  paying 
$17.10  for  an  article,  with  discounts  of  20  %,  10  %,  and  5  % 
from  the  list  price,  if  he  sells  it  at  the  list  price  ? 

2.  Find  (a)  the  ratio  of  the  areas  of  two  similar  rectangles, 
the  length  of  one  being  36  rods  and  that  of  the  other  90  rods; 
(6)  the  ratio  of  the  volumes  of  two  similar  spheres,  the  di- 
ameter of  one  being  6  feet  and  that  of  the  other  8  feet.  State 
the  principle  applied  in  each  case. 

3.  A  tank  to  hold  100  barrels  can  be  only  5  feet  wide  and 
4J  feet  deep.     What  is  the  required  length  ? 

4.  If  to  alcohol  which  cost  $  1.25  a  quart  20  %  of  its  volume 
of  water  is  added,  what  will  be  the  rate  per  cent  of  profit  if 
the  mixture  is  sold  at  $  1.40  a  quart? 

6.  If  a  certain  fraction  is  increased  by  J  of  itself,  the  result 
multiplied  by  ^  and  the  product  divided  by  ^,  the  reciprocal  of 
the  result  will  be  4^^.     Find  the  fraction. 


122  MATHEMATICAL   WRINKLES 

6.  Using  the  mercantile  rule,  find  the  amount  due  May  18, 
1907,  on  a  note  for  $  650,  given  Nov.  30,  1903,  on  which  the 
following  payments  have  been  indorsed :  Jan.  12,  1905,  $  225 ; 
April  23,  1906,  $  250.     (Use  legal  rate  of  interest.) 

7.  Determine  the  number  of  rods  around  a  square  field, 
the  diagonal  of  which  is  340  rods. 

8.  A  man  has  an  income  of  $  1925  for  an  investment  in 
United  States  Steel  stock  paying  7  %,  purchased  at  107,  bro- 
kerage I".  How  does  this  income  compare  with  that  of  the 
same  sum  invested  in  a  real  estate  mortgage  paying  5  %  ? 

9.  If  $  260  placed  at  interest  for  1  year  6  months  and  20 
days  at  6  %  produces  $  24.27  interest,  what  sum  placed  at 
interest  for  11  months  and  24  days  at  7  %  will  produce  $20 
interest  ?     (Solve  by  proportion.) 

10.  With  no  allowance  for  waste,  how  many  feet  of  lumber, 
board  measure,  will  it  take  to  make  a  watering  trough  18  feet 
long,  2^  feet  wide,  and  20  inches  deep,  outside  measurements, 
with  lumber  1^  inches  thick  ? 

County  Examination.  —  Ohio 

1.  Explain  the  meaning  of  the  following:  notation,  com- 
posite number f  insurance  premium,  commission  merchant,  trade 
discount. 

2.  If  A  cuts  21  cords  of  wood  in  7i  hours,  and  B  3^  cords 
in  8J  hours,  how  long  will  it  take  the  two  together  to  cut 
enough  wood  to  make  a  pile  170  feet  long,  4  feet  wide,  and 
6  feet  high  ? 

3.  (a)  In  the  expression  "  3  %  stock  at  75,''  explain  fully 
what  is  meant.  (6)  Make  and  solve  a  problem  to  show  clearly 
the  difference  between  true  discount  and  bank  discount. 

4.  A  person  owns  $15,000  bank  stock  paying  5  %,  which 
he  sells.     He  invests   the  proceeds  in  6  %  stock  at  120,  his 


EXAMINATION  QUESTIONS  123 

income  being  increased  $  60.     Find  the  price  at  which  he  sold 
the  first  stock. 

5.  The  side  of  a  square  inscribed  in  a  circle  is  10  feet.  Find 
both  the  diameter  and  area  of  the  circle. 

6.  A  miner  sold  2  pounds  of  gold  dust  at  $  220  a  pound 
avoirdupois,  and  the  broker  sold  it  at  $  16  per  ounce  Troy. 
Did  he  gain  or  lose,  and  how  much  ? 

7.  Write  a  rule  for  finding  the  area  of  a  rectangle,  and  illus- 
trate by  a  diagram  that  children  can  understand. 

8.  A  man  owns  a  house  valued  at  $  1500,  land  valued  at 
$  2100,  and  has  $  1500  in  a  savings  bank.  If  he  owes  $  900 
and  the  tax  rate  is  18  mills,  what  is  the  amount  of  his  tax  ? 

County  Examination.  —  Texas 

1.  There  are  two  general  methods  of  performing  subtrac- 
tion.    Explain  the  method  you  use  and  justify  its  use. 

2.  Explain  as  you  would  to  a  class  that  a  fraction  may  be 
considered  a  problem  in  division. 

3.  How  was  the  length  of  the  meter  determined?  The 
weight  of  the  gram  ?     The  capacity  of  the  liter  ? 

4.  Nine  men  can  do  a  work  in  8J  days.  How  many  days 
may  3  men  remain  away  and  yet  finish  the  work  in  the  same 
time  by  bringing  5  more  with  them  ?  r 

5.  How  many  square  inches  in  one  face  of  a  cube  which 
contains  2,571,353  cubic  inches  ? 

6.  Find  the  sum  whose  true  discount  by  simple  interest  for 
4  years  is  $  25  more  at  6  %  than  at  4  %  per  annum. 

7.  Find  the  length  of  a  minute-hand  whose  extreme  point 
moves  4  inches  in  3  minutes  28  seconds. 


124  MATHEMATICAL  WRINKLES 

8.  A,  B,  and  C  dine  on  8  loaves  of  bread ;  A  furnishes  5 
loaves ;  B,  3  loaves ;  and  C  pays  8  cents  for  his  share.  How 
must  A  and  B  divide  the  money  ? 

9.  Bought  bonds  at  12%  premium  and  sold  them  at  a  loss 
of  12i  %.     At  what  discount  were  they  sold  ? 

10.  (a)  At  what  discount  should  7%  bonds  be  bought  to 
make  8  %  on  the  investment  ? 

(6)  At  what  premium  should  8  %  bonds  be  bought  to  realize 
6|%  on  the  investment? 

Training  Class  Certificate.  —  New  York 

1.  Distinguish  between  the  simple  and  the  local  value  of  a 
figure.  How  much  greater  is  the  local  value  of  8  in  the  fourth 
order  of  units  than  in  the  second  decimal  place  ? 

2.  A  student  paid  -J-  of  his  yearly  allowance  for  books  and 
y^Q-  of  the  remainder  for  clothes ;  he  paid  $  20  more  for  clothes 
than  for  books.     What  was  his  yearly  allowance  ? 

3.  The  earth  removed  in  excavating  a  cellar  33  feet  wide 
and  55  feet  long,  to  a  depth  of  6  feet,  is  used  to  raise  the  sur- 
face of  a  lot  containing  i  of  an  acre.  How  much  is  the  surface 
of  the  lot  raised  ? 

4.  It  is  9  A.M.  at  a  place  18°  30'  east  of  New  York.  What 
is  the  time  at  a  place  46°  15'  west  of  New  York  ?  Give  a 
model  explanation. 

5.  The  net  proceeds  of  a  shipment  of  500  tons  of  hay  was 
$  6790  after  a  commission  of  3  %  had  been  deducted.  What 
was  the  selling  price  per  ton  ? 

6.  If  46%  of  the  enrollment  of  a  school  is  boys  and  there 
are  162  girls,  how  many  boys  are  enrolled  ?     Analyze. 

7.  Give  a  clear  explanation  of  the  process  of  finding,  by 
factoring,  the  lowest  common  multiple  of  78,  195,  117. 

8.  Describe  a  lesson  to  develop  the  table  of  square  measure. 


EXAMINATION  QUESTIONS  125 

For  Second  Grade  Certificate.  —  Michigan 

1.  (a)  What  is  the  least  number  by  which  |,  ^^,  and  f  can 
be  multiplied  to  give,  in  each  case,  an  integer  for  a  product? 

(6)  Divide  some  number  selected  by  yourself  into  integral 
parts  having  the  ratios  of  |,  },  and  3,  respectively. 

2.  (a)  What  is  the  volume  in  cubic  inches  of  a  body  that 
weighs  10  pounds  in  air  and  8  pounds  in  water  ? 

(b)  The  specific  gravity  of  cork  is  .24,  of  gold  is  19.36.  How 
much  gold  can  be  kept  from  sinking  by  a  cubic  foot  of  cork  ? 

3.  A  can  do  as  much  work  in  a  day  as  B  in  1^  days.  If  A 
can  do  a  piece  of  work  in  12  days,  how  long  for  them  to  do  the 
work  together  ? 

4.  Sold  two  horses  at  S  120  each.  On  one  I  lost  25  %,  on 
the  other  I  gained  enough  to  retrieve  this  loss.  What  per 
cent  did  I  gain? 

5.  When  a  certain  number  is  divided  by  45  there  is  a  re- 
mainder of  30.  What  would  be  the  remainder  if  the  number 
were  divided  by  9  ? 

6.  Give  the  following  tables,  using  proper  abbreviations: 
linear  measure,  square  measure,  liquid  measure,  and  avoirdu- 
pois weight. 

7.  Mr.  Charles  Brown  has  a  note  for  $  250  at  6  %  interest 
per  annum,  running  two  years,  which  was  given  in  Detroit  15J 
months  ago  to  John  R.  Clark  and  by  Clark  prope;*ly  indorsed 
to  Brown.  Draw  the  note,  making  proper  indorsement  and 
find  the  interest  due  to-day. 

8.  Analyze:  Ten  per  cent  of  a  consignment  of  eggs  were 
broken.  At  what  per  cent  advance  must  the  remainder  be 
sold  to  realize  a  gain  of  25  %  ? 

9.  Formulate  and  solve  an  example  in  both  simple  and  com- 
pound proportion. 


126  MATHEMATICAL   WRINKLES 

10.  Illustrate  in  a  township  the  following  described  parcel 
of  land  and  find  its  value  at  $  12.50  per  acre :  N.  i  of  N.  E.  J 
of  S.  E.  I,  sec.  16. 

11.  Define  (a)  multiple,  (b)  factor,  (c)  cancellation,  (d)  deci- 
mal fraction,  (e)  abstract  number,  (/)  ratio,  (g)  percentage, 
(h)  per  cent. 

12.  Give  principles  upon  which  the  following  operations  are 
based :  (a)  reducing  fractions  to  lower  terms,  (h)  reducing 
fractions  to  a  common  denominator,  (c)  pointing  off  in  multi- 
plication of  decimals,  {d)  dividing  percentage  by  rate  to  find 
the  base. 

13.  At  $2.50  per  rod  what  will  it  cost  to  fence  a  square 
field  containing  10  acres? 

14.  A  jobber  retails  at  a  gain  of  25%  and  discounts  this 
price  at  20%  and  10%  for  cash.  What  per  cent  are  his 
profits  on  cash  sales  ? 

Advanced  Arithmetic.  —  New  York 

1.  State  three  principles  of  the  Roman  notation  and  illus- 
trate each.  Mention  two  common  uses  of  this  system  and  two 
advantages  that  the  Arabic  system  has  over  the  Roman. 

2.  Subtract  6589  from  14,523  and  prove  the  correctness  of 
your  result  by  the  method  of  (a)  casting  out  9's,  (b)  summing 
up  the  digits  (unitate  method). 

3.  Using  the  contracted  method,  find  the  product  of 
.134567  and  8.4032  correct  to  four  places  of  decimals. 

4.  If  18  men  can  do  a  piece  of  work  in  24  days,  in  how 
many  days  can  27  men  do  the  work  ?  Solve  by  (a)  analysis, 
(6)  proportion. 

5.  If  the  price  of  milk  rises  from  6  cents  to  9  cents  a 
quart,  what  per  cent  is  the  advance?  If  the  price  falls  from 
9  cents  to  6  cents,  what  per  cent  is  the  fall  ?     Explain  in  full. 


EXAMINATION  QUESTIONS  127 

6.  A  boat  travels  15  miles  downstream  in  2Ji-  hours ;  the 
boat's  rate  of  travel  in  still  water  is  4 J  miles  au  hour.  In 
what  time  can  the  boat  return  ?     Write  analysis  in  full. 

7.  A  grocer  has  defective  scales  which  indicate  ^  ounce  less 
to  the  pound  than  the  true  weight.  What  is  the  value  of  the 
tea  that  he  sells  for  $  16.64  ?     Write  analysis  in  full. 

8.  The  exact  interest  on  a  debt  for  a  given  number  of  days 
and  at  a  given  rate  is  $9.25.  What  would  be  the  interest  on 
the  same  debt  for  the  same  time  and  at  the  same  rate  if  com- 
puted by  the  6  %  method  ?     Explain. 

Teachers'  Examination.  —  Indiana 

1.  Bought  240  barrels  of  apples  at  $1.75  a  barrel ;  lost  40 
barrels  through  frost.  At  what  price  a  barrel  must  I  sell  the 
remainder  to  gain  25  %  on  the  money  invested  ? 

2.  Find  cost  of  stone  wall  4  rods  long,  6  feet  high,  and  2  feet 
thick,  at  60  ^  a  square  foot. 

3.  Simplify  the  following: 

3i  +  2^-H^1.375. 

4.  A  resident  of  the  city,  giving  up  his  lease  on  a  house  at 
$30  per  month,  bought  a  lot  at  S 1200  and  built  a  house  costing 
$  2400.  Taxes  per  year  are  $  56.70 ;  cost  of  insurance  $  10,  and 
cost  of  repairs  S  25.  Allowing  interest  at  6  %  on  the  amount 
in  the  property,  how  much  does  he  save  annually  by  owning 
his  own  property  ? 

5.  After  wheeling  12^  miles,  a  boy  found  he  had  traveled 
83^  %  of  the  distance  he  had  intended  to  go.  How  long  a 
ride  did  he  expect  to  take  ? 

6.  The  wheels  of  a  locomotive  are  15  feet  6  inches  in  cir- 
cumference and  make  8  revolutions  a  second.  How  long  does 
it  take  it  to  run  100  miles  ? 


128  MATHEMATICAL   WEINKLES 

7.  Central  Park,  New  York,  contains  879  acres,  and  the  new- 
reservoir  in  the  Park  contains  107  acres.  What  per  cent  of 
the  park  does  the  reservoir  cover  ? 

8.  Find  the  interest  on  $  1150  for  1  year  3  months  and  17 
days  at  6%. 

County  Examination.  —  Texas 

1.  Three  boys  had  169  apples  which  they  shared  in  the 
ratio  of  ^,  ^,  and  ^.     How  many  did  each  receive  ? 

2.  What  is  the  difference  in  area  between  a  half  of  a  foot 
square  and  half  of  a  square  foot  ? 

3.  A  man  living  in  Galveston-  observed  that  his  clock,  cor- 
rect by  sun  time,  was  19  minutes  slower  than  the  depot  clock, 
correct  by  standard  time,  90th  meridian.  Eind  longitude  of 
Galveston. 

4.  A  merchant  bought  cloth  at  $1.15  per  meter  and  sold 
it  by  the  yard  at  a  profit  of  20  % .  How  much  did  he  get  per 
yard? 

5.  The  distance  from  Austin  to  San  Antonio  is  152,064 
varas.     Find  the  distance  in  miles. 

6.  A  merchant  paid  $1323  for  goods,  and  the  discounts 
were  25  %,  121  %,  and  10  %.     Find  the  list  price. 

7.  An  agent  sells  1200  barrels  of  apples  at  $4.50  a  barrel 
and  charges  2^%  commission.  After  deducting  his  commis- 
sion of  8  %  for  buying,  he  invests  the  net  proceeds  in  cotton. 
What  is  his  entire  commission  ? 

8.  How  much  must  be  invested,  if  stock  20  %  below  par 
yield  a  6  %  income  of  $  390  ? 

9.  How  large  a  draft,  payable  in  30  days  after  sight,  can 
be  bought  for  $  352.62,  exchange  li  %  discount,  and  interest 
at  6  %  ? 


EXAMINATION  QUESTIONS  129 

10.  A  grocer  has  a  false  balance  which  gives  14^  ounces  to 
the  pound.  What  does  he  gain  by  the  cheat  in  selling  sugar 
for  $258.56? 

11.  What  would  be  the  cost  of  10  planks  each  18  feet  long, 
15  inches  wide,  2  inches  thick,  at  $40  per  thousand  board  feet? 

For  State  Certificate.  —  Ohio 

1.  A  and  B  run  a  race,  their  rates  of  running  being  as  17  to 
18.  A  runs  2J  miles  in  16  minutes  48  seconds,  and  B  the 
whole  distance  in  34  minutes.     What  is  the  distance  run  ? 

2.  The  surface  of  the  six  equal  faces  of  a  cube  is  1350 
square  inches.    What  is  the  length  of  the  diagonal  of  the  cube  ? 

3.  A  man  bought  5  %  stock  at  109^,  and  4J  %  pike  stock 
at  1074 ,  brokerage  in  each  case  ^  %  ;  the  former  cost  him  $  200 
less  than  the  latter,  but  yielded  the  same  income.  Find  the 
cost  of  the  pike  stock. 

4.  A,  B,  and  C  start  together  and  walk  around  a  circle  in 
the  same  direction.  It  takes  A  -^  hours,  B  |  hours,  C  |-f  hours 
to  walk  once  around  the  circle.  How  many  times  will  each 
go  around  the  circle  before  they  will  all  be  together  at  the 
starting  point  ? 

6.  I  hold  two  notes,  each  due  in  two  years,  the  aggregate 
face  value  of  which  is  S  1020.  By  discounting  both  at  5  %, 
one  by  bank,  the  other  by  true  discount,  the  proceeds  will  be 
$  923.     Find  face  of  bank  note. 

6.  The  hour  and  the  minute  hands  of  a  watch  are  together 
at  12  o'clock.     When  are  they  together  again  ? 

7.  How  many  cannon  balls  12  inches  in  diameter  can  be 
put  into  a  cubical  vessel  4  feet  on  a  side ;  and  how  many  gal- 
lons of  wine  will  it  contain  after  it  is  filled  with  the  balls, 
allowing  the  balls  to  be  hollow,  the  hollow  being  6  inches  in 
diameter,  and  the  opening  leading  tq  it  containing  1  cubic  inch  ? 


130  MATHEMATICAL   WRINKLES 

8.  An  agent  sold  a  house  at  2  %  commission.  He  invested 
the  proceeds  in  city  lots  at  3  %  commission.  His  commissions 
amounted  to  $  350.     For  what  was  the  house  sold  ? 

For  State  Certificate.  —  Tennessee 

1.  What  is  the  difference  between  common  and  decimal 
fractions  ? 

2.  Multiply  one  tenth  by  twenty-five  ten-thousandths,  di- 
vide the  product  by  five  millionth s,  and  subtract  nine  tenths 
from  the  quotient. 

3.  When  it  is  10  o'clock  a.m.  at  Berlin,  13°  23'  43"  E.,  what 
is  the  time  at  Boston,  71°  3'  30"  W.  ? 

4.  A,  B,  and  C  can  together  mow  a  field  in  25  days ;  A  can 
mow  it  alone  in  70  days,  and  B  in  80  days.  In  what  time  can 
C  mow  it  alone  ? 

5.  How  many  gallons  of  water  will  a  cistern  5  feet  in  diam- 
eter and  10  feet  in  depth  hold  ? 

6.  A  merchant  sold  a  watch  for  $  40  and  lost  20  % .  With 
the  $40  he  bought  another  watch,  which  he  sold  at  a  gain  of 
20  % .  What  was  the  merchant's  gain  or  loss  by  the  transac- 
tions ? 

7.  Find  the  annual  interest  on  $  560  for  4  years  3  months 
and  18  days. 

8.  If  1800  men  have  provisions  to  last  41  months,  at  the 
rate  of  1  pound  4  ounces  a  day  to  each,  how  long  will  five  times 
as  much  last  3500  men,  at  the  rate  of  12  ounces  a  day  to  each 
man?     (Solve  by  proportion.) 

9.  What  will  it  cost,  at  90  cents  per  yard,  to  carpet  a  room 
19  X  14^  feet,  strips  running  lengthwise,  with  carpeting  |  yard 
wide  ? 

10.   How  many  posts,  placing  them  8  feet  apart,  will  be 
required  to  fence  a  square  field  containing  16  acres  ? 


EXAMINATION  QUESTIONS  131 

State  Examination.  —  Ohio 

1.  What  fraction  is  |  of  its  reciprocal  ? 

2.  The  hands  of  a  clock  coincide  every  66  minutes.  How 
much  does  the  clock  gain  or  lose  in  one  hour  ? 

3.  Wishing  to  know  the  height  of  a  certain  steeple,  I  meas- 
ured the  shadow  of  the  same  on  a  horizontal  plane  27|^  feet. 
I  then  erected  a  10-foot  pole  on  the  same  plane  and  it  cast  a 
shadow  2|  feet.    What  was  the  height  of  the  steeple  ? 

4.  A  offered  me  a  bill  of  sugar  for  $1800  on  6  months' 
credit,  or  for  the  present  worth  of  that  sum  for  cash.  I  ac- 
cepted the  latter  offer  and  obtained  the  money  at  a  bank  for 
the  same  time  at  6  %.     Did  I  lose  or  gain  and  how  much  ? 

5.  A  stone  was  thrown  into  an  empty  cylindrical  vessel, 
which  was  then  tilled  with  water ;  when  the  stone  was  taken 
out,  the  water  fell  4.75  inches.  What  was  the  volume  of  the 
stone,  the  diameter  of  the  vessel  being  9  inches  ? 

6.  A  passenger  train  leaves  a  certain  station  at  2  o'clock,  to 
go  to  the  end  of  the  road,  120  miles,  and  travels  at  the  rate  of 
25  miles  an  hour.  At  what  time  must  a  freight  train  which 
travels  at  the  rate  of  15  miles  in  50  minutes,  have  left,  so  as 
not  to  be  overtaken  by  the  passenger  train  ? 

7.  A  owns  a  house  which  rents  for  $  1450,  and  the  tax  on 
which  is  2|%  on  a  valuation  of  $8500.  He  sells  for 
$  15,300  and  invests  in  stock  at  90  that  pays  7  %  dividends. 
Is  his  yearly  income  increased  or  diminished,  and  how  much  ? 

8.  The  distance  between  the  centers  of  two  wheels  is  12 
feet.  If  their  radii  are  7  feet  and  1  foot,  find  the  length  of  the 
belting  necessary  for  one  to  run  the  other. 

For  State  Certificate.  —  Tennessee 

1.  State  the  difference  between  common  and  decimal  frac- 
tions. 


132  MATHEMATICAL  WRINKLES 

2.  Approximately  the  longitude  of  Carthage  is  10  degrees 
15  minutes  and  20  seconds  east,  while  that  of  Colon  is  79 
degrees  25  minutes  and  30  seconds  west.  When  it  is  9  o'clock 
A.M.  at  Carthage,  what  is  the  hour  at  Colon  ? 

3.  87  %  of  961  is  29  %  of  what  number  ? 

4.  Make  formulae  for  each  case  of  percentage. 

5.  A  boy  had  two  goats  which  he  sold  for  $6  each. 
What  did  they  cost  him  if  he  gained  20  %  on  one  and  lost 
20  %  on  the  other  ? 

6.  Write  a  negotiable  promissory  note ;  a  draft ;  a  check. 

7.  Find  the  annual  interest  on  $760  at  5  per  cent  for 
4  years  5  months  18  days. 

How  long  must  $84.80  be  put  on  interest  at  5^%  to  amount 
to  $102.29? 

8.  Divide  65  into  parts  proportional  to  ^,  ^,  and  \. 

9.  If  a  mow  of  hay  32  feet  long,  16  feet  wide,  and  16  feet 
high  lasts  8  horses  20  weeks,  how  many  weeks  will  a  mow  of 
hay  28  feet  long,  20  feet  wide,  and  12  feet  high  last  5  horses? 

10.   Two  trees,  80  and  120  feet  high,  respectively,  are  30 
yards  apart.     What  is  the  distance  between  their  tops  ? 

Teachers'  Examination.  —  Ohio 

1.  Find  the  decimal  which  when  added  to  the  difference 
of  ^^  and  0.002775  produces  the  square  of  0.215. 

2.  A  can  do  a  piece  of  work  in  2  hours,  B  in  21  hours,  and 
C  in  3i  hours.  How  much  of  the  work  can  they  do  in  20 
minutes,  all  working  together  ? 

3.  Find  the  principal  that  will  amount  to  $131.88  in  2 
years  11  months  15  days  at  6  %. 

4.  Write  an  example  in  trade  discount  and  give  solution. 


EXAMINATION  QUESTIONS  133 

5.  Sold  an  invoice  of  books  at  a  loss  of  16J%.  Had  I 
paid  $400  less,  my  gain  would  have  been  25%.  What  was 
the  selling  price  ? 

6.  A's  money  added  to  |  of  B's,  which  is  to  A^s  as  2  is  to 
3,  being  put  on  interest  for  6  years  at  4  %  amounts  to  $744. 
How  much  money  has  each  ? 

7.  I  received  $  4850  and  a  consignment  of  2000  barrels  of 
flour  which  I  sold  at  $7.50  a  barrel  and  invested  the  net 
proceeds  and  cash  in  cotton.  How  much  did  I  invest  in  cotton, 
my  commission  being  3%  for  selling  and  1^%  for  buying, 
and  the  expenses  for  storage  and  freight  S350? 

8.  What  should  be  paid  for  a  6  %  stock  that  8  %  may  be 
realized  on  the  investment  ? 

9.  When  do  the  hour  and  minute  hand  of  a  watch  coincide 
between  8  and  9  o'clock  ? 

10.  A  bushel  measure  and  a  peck  measure  are  of  the  same 
shape.     Find  the  ratio  of  their  heights. 

For  County  Superintendent.  —  Tennessee 

1.  A  man  has  1\  miles  to  go ;  after  he  has  gone 

i  +  ^xH-l 
fxlj-hl 

of  a  mile,  how  far  has  he  yet  to  go  ? 

2.  Simplify:   (0.08J4- 1.2i)-5-(0.006J  x  0.016). 

3.  Reduce  2  pecks  3  quarts  1.2  pints  to  the  decimal  of  a 
bushel. 

4.  A  man  sold  two  horses  for  $  200  each.  On  one  he  made 
50  %  of  the  cost,  and  on  the  other  he  lost  50  %.  Did  he  make 
or  lose  by  these  sales,  and  how  much  ? 

5.  A  merchant  sends  his  agent  $  10,246.50  with  which  to 
buy  flour.  After  deducting  his  commission  of  3^  %,  how  many 
barrels  of  flour  at  $  5.50  a  barrel  can  be  purchased  ? 


134  MATHEMATICAL   WRINKLES 

6.  A  note  of  $  850  with  interest  payable  annually  at  5  % 
was  paid  3  years  3  months  18  days  after  date,  and  no  interest 
had  previously  been  paid.     What  was  the  amount  due  ? 

7.  What  is  the  exact  interest  on  $  600  at  5  %  for  90  days  ? 

8.  If  4  men  can  dig  a  ditch  72  rods  long,  5  feet  wide,  and 
2  feet  deep  in  12  days,  how  many  men  can  dig  a  ditch  120  rods 
long  6  feet  wide  1  foot  6  inches  deep  in  9  days  ? 

9.  Find  the  cube  root  of  28.094464. 

10.   A  man  receives  $  630  as  his  annual  dividend  from  7  % 
stock.     How  many  shares  of  $100  each  does  he  hold. 

Teachers'  Examination.  —  Georgia 

1.  What  is  a  Unit?  A  Number?  A  pure,  or  abstract, 
Number  ?     What  is  an  Integer  ? 

2.  At  what  time  should  Wentworth's  Elementary  Arith- 
metic be  taken  up  ?  What  kind  of  training  ought  the  child 
to  have  had  as  an  introduction  to  book  work  ? 

3.  What  powers  ought  to  receive  special  training  before 
book  work  is  begun  ? 

4.  Give  suggestions  of  lessons  intended  to  train  (a)  the 
eye,  (b)  the  ear,  (c)  the  touch.  Would  any  good  purposes  be 
served  by  having  arithmetic  lessons  relate  generally  to  the 
community  and  its  life  ?     Why  ? 

5.  Change  the  following  numbers  in  Roman  Notation  into 
Arabic  Notation : 

DXLVI,  MCDXCII,  CCIV,  MDCCCCXI,  DCXL 

6.  Define  the  following:  a  Prime  Number;  a  Composite 
Number ;  Factor ;  Multiple ;  Least  Common  Multiple. 

7.  A  farmer  who  owned  f  of  an  acre  of  land  sold  f  of  his 
share  at  the  rate  of  $300  an  acre.  How  much  did  he  get 
for  it? 


EXAMINATION  QUESTIONS  135 

8.  What  is  Ratio  ?  Proportion  ?  The  Washington  Monu- 
ment casts  a  shadow  223  feet  6^  inches  when  a  post  3  feet  high 
casts  a  shadow  14.5  inches.  What  is  the  height  of  the  monu- 
ment? 

9.  A  man  bought  20  acres  of  land  at  $50.25  an  acre.  He 
sold  i  of  an  acre  to  B,  8|  acres  to  C,  and  the  remainder  to  D. 
If  he  received  $65  an  acre  from  B  and  C,  and  $60  an  acre 
from  D,  how  much  did  he  gain  ? 

10.   James  McKnight  bought  from  James  Laird,  Charleston, 
S.C.,  as  follows : 

40  joists  2  X  6,  18  feet  long,  at  $  25  per  M. 
16  beams  6  x  9,  20  feet  long,  at  $30  per  M. 
72  scantling  2  x  4,  12  feet  long,  at  $  24  per  M. 
240  boards  1  x  10,  12  feet  long,  at  $  18  per  M. 
24  planks  2  x  14,  16  feet  long,  at  $17.50  per  M. 
Make  out  complete  bill,  and  find  amount  due  Laird. 

For  County  Certificate.  —  Louisiana 

1.  Find  the  difference  between  IJ  x  2|  and  0.019  of  220. 

2.  Express  ratio  of  25|  yards  to  14}  rods  in  three  different 
ways :  first,  as  a  common  fraction  in  its  lowest  terms ;  second, 
as  a  decimal  fraction ;  and,  third,  a  rate  per  cent. 

3.  On  November  21,  1908,  Henry  Brown  loaned  to  Peter 
White  on  his  note  for  2  years  at  8  per  cent,  $500. 

Write  the  note.    Payments  on  the  note  were  made  as  follows : 

Jan.  1,  1909 $200 

Sept.  15,  1909 125 

What  was  due  at  maturity  of  note  ? 

4.  A  real  estate  dealer  asked  for  a  farm  25  per  cent  more 
than  it  cost.  He  finally  took  15  per  cent  less  than  the  asking 
price  and  gained  $  1000.  What  was  his  asking  price  ?  (Ana- 
lyze.) 


136  MATHEMATICAL   WKINKLES 

5.  If  4  men  dig  a  trench  in  15  days  of  10  hours  each,  in 
how  many  days  of  8  hours  each  can  5  men  perform  the  same 
work  ?     (Analyze.) 

6.  What  will  be  the  cost  of  a  pile  of  wood  20  feet  x  14  feet 
X  12  feet  at  $3.50  a  cord? 

7.  A,  B,  and  C  enter  into  partnership.  A  puts  in  $500  for 
5  months,  B  puts  in  $1000  for  8  months,  and  C  $1500  for  2 
years.     They  gain  $1200.     What  is  the  share  of  each  ? 

Teachers'  Certificate.  —  Florida 

1.  A  man  having  100  fowls  sold  J  of  them  to  E  and  |  of 
the  remainder  to  F.  What  was  the  value  of  what  remained,  if 
they  were  worth  26  cents  apiece  ? 

2.  What  is  the  exact  value  of  /^3  +  2i  -  f  of  f  +  -  V  4^  ? 


3.  A  man  sold  8  bushels  3  pecks  4  quarts  of  cranberries  at 
$3|-  a  bushel,  and  took  his  pay  in  flour  at  3^  cents  a  pound. 
How  many  barrels  of  flour  did  he  receive  ? 

4.  The  difference  in  time  between  London  and  New  York 
is  4  hours  55  minutes  37f  seconds.  What  is  their  difference 
in  longitude  ? 

5.  How  much  less  would  it  cost  to  make  a  brick  sidewalk 
41  feet  wide  and  260  feet  long,  at  $  1.08  a  square  yard,  than  to 
lay  a  stone  walk  of  the  same  dimensions,  at  22  cents  a  square 
foot? 

6.  A  merchant  marked  cloth  at  25  %  advance  on  the  cost. 
The  goods  being  damaged,  he  was  obliged  to  take  off  20  %  of 
the  marked  price,  selling  it  at  $1  per  yard.  What  was  the 
cost? 

7.  What  is  the  duty  on  18  pieces  of  Brussels  carpeting,  of 
60  yards  each,  invoiced  at  45  cents  per  yard,  the  specific  duty 
being  38  cents  per  yard,  and  the  ad  valorem  duty  35  %  ? 


EXAMINATION  QUESTIONS  137 

8.  If  9  men  can  mow  75  acres  of  grass  in  6  days  of  8 J 
hours  each,  in  how  many  days  of  8  hours  each  can  15  men  mow 
198  acres  ? 

9.  A  merchant  bought  a  bill  of  goods  amounting  to  $3257 
on  a  credit  of  3  months,  but  was  offered  a  discount  of  2^  %  for 
cash.  How  much  would  he  have  gained  by  paying  cash,  money 
being  worth  7  %  ? 

10.    How   many  cubic  feet  are  there  in  a  spherical  body 
whose  diameter  is  25  feet  ? 

Teachers'  Examination.  —  California 

Orcd  Arithmetic 

* 

1.  I  sold  a  horse  for  S  60  and  thereby  lost  J  of  the  cost. 
What  should  I  have  sold  it  for  to  gain  ^  of  the  cost  ? 

2.  If  to  a  certain  number  ^  of  itself  and  J  of  itself  be 
added,  the  sum  will  be  66.     Find  the  number. 

3.  A  bicyclist  rode  27  miles  in  2  hours  15  minutes.  What 
was  the  rate  in  miles  per  hour  ? 

4.  What  is  the  squai'e  of  3^?  Answer  to  be  a  mixed 
number. 

5.  Write  equivalent  common  fractions  for  the  following 
decimals :  .87^,  .62^,  .06^. 

6.  A,  B,  and  C  enter  into  partnership.  A  puts  in  $  400  for 
1  year;  B  $300  for  2  years;  C  $200  for  4  years;  they  gain 
$720.     What  is  the  share  of  each  ? 

7.  Sold  24  boxes  of  apples  at  $1.50  a  box,  and  bought  cloth 
with  the  proceeds  at  $  .75  a  yard.     How  many  yards  did  I  buy  ? 

8.  ^hatpercentof  51iis  174? 

9.  A  field  containing  3200  square  rods  is  just  twice  as  long 
as  it  is  wide.     What  are  its  dimensions  ? 

10.  3-!-ixi  +  2J-6i8  J  of  what  number? 


138  MATHEMATICAL   WRINKLES 

For  State  Certificate.  —  Washington 

1.  Analyze:  A  has  20%  more  money  than  B,  who  has 
25  %  more  than  C.  A  has  $80  more  than  C.  How  much  has 
each  ? 

2.  Analyze:  A  can  do  a  piece  of  work  in  13  days,  B  in  18 
days,  and  C  in  20  days.  After  all  have  worked  4  days,  how 
long  will  it  take  C,  working  alone,  to  finish  ? 

3.  If  the  proceeds  of  a  sale  of  20  tons  of  potatoes,  allow- 
ing 4%  commission,  was  $432,  at  what  price  per  hundred- 
weight were  they  sold? 

4.  Goods  marked  to  be  sold  at  35  %  profit,  were  sold  at  a 
discount  of  20  %  from  marked  price ;  the  gain  was  $  192. 
AVhat  was  the  marked  price? 

5.  What  is  the  capacity  in  liters  of  a  tank  4  meters  6  deci- 
meters long,  3  meters  2  decimeters  wide,  2  meters  5  decimeters 
deep  ?     What  is  the  capacity  in  kiloliters  ? 

6.  Principal  $  675 ;  time  1  year  6  months.  Find  amount 
and  write  the  note  in  full,  making  it  negotiable  by  indorsement. 

7.  Find  one  edge  of  a  cube  whose  volume  is  2515.456  cubic 
inches. 

8.  If  24  men  in  15  days  of  12  hours  each  dig  a  trench  300 
rods  long,  5  yards  wide,  and  6  feet  deep,  in  how  many  days  of 
10  hours  each  can  45  men  dig  a  trench  125  rods  long,  5  yards 
wide,  and  8  feet  deep  ?     (Solve  by  proportion.) 

9.  (a)  Find  f  of  3  miles  64  rods  3  yards  2  feet  8  inches. 
(6)  Express  .45  mile  in  integers  of  lower  denominations. 

10.  Find  the  number  of  board  feet  in  four  pieces  10"  x  2'  x  16', 
two  pieces  10"  x  8"  x  32',  and  one  piece  12"  x  12"  x  40'. 

11.  Find  the  volume  of  the  largest  square  prism  that  can  be 
cut  from  a  cylinder  4  feet  in  diameter,  12  feet  long. 


EXAMINATION  QUESTIONS  139 

For  State  Certificate.  —  Washington 

1.  Analyze :  A  horse  cost  one  fourth  more  than  a  carriage ; 
the  horse  was  sold  for  20  %  more  than  cost,  and  the  carriage 
for  20%  less  than  cost.  Both  together  sold  for  $368.  What 
was  the  cost  of  each  ? 

2.  Analyze :  At  what  time  between  8  and  9  o'clock  are  the 
hands  of  a  watch  together  ? 

3.  When  it  is  6  p.m.  at  St.  Paul  95°  4'  55"  west,  it  is  33 
minutes  54  seconds  after  1  a.m.  next  day  at  Constantinople. 
What  is  the  longitude  of  Constantinople  ? 

4.  Find  the  proceeds  of  note  of  S825,  drawing  interest  at 
7  %  per  annum,  given  April  25,  1908,  due  6  months  after  date, 
discounted  July  13  at  8  %  per  annum. 

5.  What  annual  income  is  derived  from  $8475  invested  in 
5^  %  bonds  bought  at  113  ? 

6.  (a)  What  number  is  40  %  more  than  850  ? 

(6)  1050  is  how  many  per  cent  more  than  630  ? 
(c)  What  number  is  20  %  less  than  800  ? 
{d)  600  is  25  %  less  than  what  number  ? 
(e)  900  is  how  many  per  cent  less  than  1200  ? 

7.  The  hypotenuse  of  a  right  triangle  is  115,  its  altitude  is 
92.     What  is  its  base  ?    What  is  its  area? 

8.  The  dimensions  of  a  rectangular  solid  are  24  inches,  20J 
inches,  and  12  inches.  Find  its  area  and  volume.  Find  the 
edge  of  a  cube  of  equal  volume. 

9.  Find  the  area  in  hectares  of  a  field  30  dekameters  in 
length,  20  dekameters  in  width. 

10.  If  the  freight  on  30  head  of  cattle,  each  weighing  1400 
pounds,  for  a  distance  of  160  miles,  is  $  112,  what  should  be 
the  freight  on  36  head,  each  weighing  1800  pounds,  for  a  dis- 
tance of  140  miles  ?     (Solve  by  proportion.) 


140  MATHEMATICAL   WEINKLES 

For  State  Certificate.  —  Oregon 

1.  A  well  at  Madison,  Wisconsin,  furnishes  enough  water 
to  irrigate  110  acres  of  land  2  inches  deep,  every  10  minutes. 
At  this  rate  how  many  acres  can  it  cover  to  the  depth  of  1  inch 
every  day  ? 

2.  A  dealer  bought  two  horses  at  the  same  price.  He  sold 
one  at  a  profit  of  20  %  for  $  102.  The  other  he  sold  at  a  loss 
of  10%.     How  much  did  he  receive  for  the  latter? 

3.  (a)  Find  the  interest  on  S  625.20  for  6  months  9  days 
at  5  % .  (b)  Some  4-foot  wood  is  piled  5  feet  high.  The  pile 
is  2  rods  long.     How  many  cords  are  there  ? 

4.  Find  the  discount  and  proceeds  of  the  following  note: 
Face,  $175.     Time,  four  months  without  grace.     Eate,  6%. 

5.  An  agent  has  $590  to  invest  after  deducting  his  com- 
mission of  2  %  on  the  money  invested.  What  amount  does  he 
invest  ? 

6.  The  distance  around  a  square  farm  is  3  miles  240  rods. 
Find  the  length  of  each  side ;  the  area  in  acres. 

7.  Allowing  231  cubic  inches  to  the  gallon,  how  many  gal- 
lons in  a  watering  trough  that  is  6  feet  long  and  16  inches 
wide,  the  ratio  of  its  depth  to  its  width  being  3:4? 

8.  A  boy  in  a  grocery  store  receives  $  8  a  week.  He  spends 
20%  of  it  for  board,  20%  of  the  remainder  for  clothes,  and 
$2  in  other  ways.  If  he  saves  the  rest,  how  much  will  he 
save  in  a  year  ? 

9.  (By  proportion.)  When  2  men  can  mow  16  acres  of 
grass  in  10  days,  working  8  hours  a  day,  how  many  men 
would  it  take  to  mow  27  acres  in  9  days,  working  10  hours  a 
day? 

10.   How  long  must  a  pile  of  wood  be  to  contain  10  steres, 
if  it  is  3.5  meters  high  and  3.8  meters  wide  ? 


EXAMINATION  QUESTIONS  141 

11.  The  diagonal  of  one  face  of  a  cube  is   V162  inches. 
Find  the  surface  and  the  volume  of  the  cube. 

12.  What  will  it  cost  to  gild  a  ball  25  inches  in  diameter  at 
$13.50  a  square  foot? 

Examination  for  Teachers'  Certificate.  —  Pennsylvania 

1.  The  longitude  of  Washington,  D.C.,  is  77°  03'  06"  west. 
Tokyo  is  139°  44'  30"  east.  When  it  is  6  o'clock  p.m.  in 
Washington,  Feb.  10,  what  is  the  time  in  Tokyo  ? 

2.  How  many  yards  of  carpet  27  inches  wide  are  required 
to  cover  a  floor  20  feet  long  and  15  feet  wide,  allowing  5^ 
yards  for  matching? 

3.  On  March  9,  1908,  John  Doe  bought  a  house  from 
Richard  Roe  for  $6000;  20%  of  the  price  was  paid  immedi- 
ately and  a  6-niouths  note  bearing  6%  interest,  given  for  the 
remainder.  The  note  was  discounted  at  bank  April  9.  Write 
the  note  and  find  the  discount. 

4.  Three  contractors,  A,  B,  and  C,  did  work  for  which  they 
received  $1500.  A  furnished  12  men  24  days;  B,  20  men  12 
days ;  and  C,  18  men  20  days.     What  is  the  share  of  each  ? 

6.  How  far  is  it  between  the  tops  of  two  trees  which  are 
80  feet  apart,  if  their  heights  are  40  feet  and  100  feet  respec- 
tively ? 

6.  The  weight  of  a  ball  4  inches  in  diameter  was  8  pounds ; 
\  of  the  diameter  was  turned  off.  How  many  cubic  inches 
were  turned  oft",  and  what  was  its  weight  then  ? 

Teachers'  Examination.  —  Washington 

1  A  boy,  after  doing  f  of  a  piece  of  work  in  30  days,  is 
assisted  by  his  father,  with  whom  he  completes  the  work  in 
6  days.  How  long  would  it  have  taken  each  to  do  the  work 
alone  ?     (Analyze  in  full.) 


142  MATHEMATICAL  WRINKLES 

2.  A  fruit  dealer  bought  oranges  at  the  rate  of  40  for  $1, 
and  sold  them  at  50  cents  per  dozen.  Find  gain  per  cent.  He 
also  bought  apples  at  the  rate  of  5  for  2  cents  and  sold  them  at 
8  cents  per  dozen.  How  many  must  he  buy  and  sell  in  order  to 
gain  $2?     (Analyze  in  full.) 

3.  A  farmer  finds  that  a  bin  8  feet  long,  3  feet  6  inches  wide, 
and  5  feet  deep  holds  about  112  bushels.  How  many  bushels 
may  be  contained  in  a  bin  50  %  longer,  twice  as  wide,  and  50  % 
as  deep  ? 

4.  A  man  who  owns  a  quarter  section  of  coal  land  claims 
that  he  has  a  bed  of  coal  6  feet  thick  covering  the  entire 
quarter.  If  so,  how  many  tons  of  coal  has  he,  allowing  40 
cubic  feet  to  the  ton  ? 

5.  The  steeple  of  a  certain  church  is  a  pyramid  28  feet  in 
slant  height  and  stands  upon  a  base  14  feet  square.  Find 
cost  of  painting  it  at  10  cents  per  square  yard. 

6.  A  certain  city  bought  two  horses  for  the  fire  depart- 
ment, but  finding  them  unfit  for  the  work,  sold  them  for  $300 
each,  thus  gaining  20  %  on  one,  and  losing  20  %  on  the  other. 
Did  the  city  gain  or  lose,  and  how  much  ? 

7.  A  rectangular  lot  contains  one  acre  and  has  a  street 
frontage  of  120  feet.  How  deep  is  the  lot  and  how  many  yards 
of  fence  are  required  to  inclose  it  ? 

8.  The  hour  hand  of  a  clock  is  4  inches  long.  Over  what 
area  does  it  pass  upon  the  dial  during  a  school  day ;  that  is, 
from  9  A.M.  to  4  p.m.? 

9.  Our  most  expensive  battleship,  the  Connecticut,  cost 
$  6,000,000,  and  the  Louisiana  97|  %  as  much.  This  was  117  % 
of  the  cost  of  the  Vermont,  which  cost  \  more  than  the  Kansas. 
Find  total  cost  of  this  division  of  our  fleet. 

10.   The  cruiser  Olympia  is  21^9  %  faster  than  the  battleship 
Oregon,  which  is  a  19-knot  vessel.     If  each  runs  at  full  speed. 


EXAMINATION  QUESTIONS  143 

how  much  can  the  former  gain  upon  the  latter  in  going  from 
Tacoma  to  Seattle,  the  distance  being  23  knots  ? 

11.  A  swimming  tank  is  40  meters  long  and  15  meters  wide. 
When  filled  to  an  average  depth  of  2  meters,  how  many  liters 
of  water  does  it  contain  ?  Find  weight  of  the  water  in  kilo- 
grams. 

For  State  Certificate.  —  Colorado 

1.  A  man  bought  a  lot  for  $1200,  and  built  a  house  for 
$1980.  He  insured  the  house  for  |  of  its  value  at  f  %.  The 
house  burned  and  the  lot  was  sold  for  $  1328.  How  much  was 
the  gain  or  loss? 

2.  At  what  price  must  you  mark  a  hat  costing  $1.50  so 
you  can  discount  the  price  20  %  and  still  make  12  %  ? 

3.  In  a  school  J  of  the  pupils  study  grammar,  |  arithmetic, 
J  geography,  and  the  remainder,  which  is  39,  write.  How 
many  pupils  in  the  school  ? 

4.  (a)  How  many  yards  of  brussels  carpet  J  yard  wide  will 
cover  a  floor  24  feet  9  inches  long  and  17^  feet  wide,  if  the 
strips  run  lengthwise  and  the  matching  of  the  figure  requires 
that  6  inches  be  turned  under  ?  (6)  What  will  the  carpet  cost 
at  S1.65  per  yard? 

5.  When  5  %  bonds  are  quoted  at  104,  what  sum  must  be 
invested  to  yield  an  annual  income  of  $  800  ? 

6.  If  14  persons  spend  $1120  in  8  months,  at  the  same 
rate,  what  will  9  of  the  same  persons  spend  in  5  months  ? 

7.  $500.00  Denver,  Colo.,  May  12,  1908. 
Ninety  days  after  date,  I  promise  to  pay  to  the  order  of 

Charles  Taylor,  Five  Hundred  Dollars,  at  the  Central  National 
Bank.    Value  received. 

John  J.  Smith. 

Discounted  May  26,  1908.    Find  the  proceeds.    Rate  6  %. 


144  MATHEMATICAL   WRINKLES 

8.  How  many  square  feet  of  surface  in  a  stovepipe  16  feet 

long  and  7  inches  in  diameter  ? 

9.  A,  B,  and  C  can  do  a  piece  of  work  in  10  days,  and  B  and 
C  can  do  it  in  18  days.     In  what  time  can  A  do  it  alone  ? 

10.  How  many  board  feet  in  16  pieces  of  lumber,  each  being 
14  feet  long,  16  inches  wide,  and  li  inches  thick  ? 

11.  The  specific  gravity  of  sand  is  3^.     How  much  will  a 
cubic  yard  of  sand  weigh  ? 

State  Examination. — Maine 

1.  What  are  fractions  ?  What  names  are  given  to  the 
terms  of  a  fraction  ?  Why  are  they  so  named  ?  What  is  the 
value  of  a  fraction  ? 

2.  Why  is  it  necessary  to  teach  L.  C.  M.  and  G.  C.  D.  before 
teaching  fractions  ?  What  two  things  must  be  taught  before 
teaching  L.  C.  M.  and  G.  C.  D.  ?  Find  the  L.  C.  M.  and  G.  C.  D. 
of  9,  12,  and  54. 

3.  Add  five-sixths,  two-fifths,  and  four-fifteenths.  State  the 
four  steps  taken  and  give  reasons  for  each. 

4.  Change  5  shillings  and  8  pence  to  the  decimal  of  a 
pound.  Write  the  tables  of  Long  and  Liquid  measures  as 
used  to-day.  How  many  cords  in  a  pile  of  wood  18  feet  long, 
4  feet  wide,  and  5i  feet  high  ?  Write  and  solve  a  problem  in 
Reduction  descending. 

5.  A  sold  B  a  farm  for  $  2400,  which  was  20  %  more  than 
it  cost  him,  and  took  B's  note  for  that  amount  due  in  6  months 
without  interest.  If  he  had  that  note  discounted  at  a  Maine 
bank,  what  was  his  actual  gain  and  what  per  cent  did  he  gain  ? 
Write  the  note  taken.  What  did  A  have  to  do  before  the  bank 
would  discount  the  note  ? 


EXAMINATION  QUESTIONS  145 

For  State  Certificate.  —  California 

1.  I  pay  $275  for  a  lot  and  build  on.it  a  house  costing 
$1720,  which  my  agent  rents  for  $25  a  month,  charging  5  % 
commission.  What  per  cent  do  I  make  on  the  money  in- 
vested? 

2.  A  house  valued  at  $1200  had  been  insured  for  |  of  its 
value  for  3  years  at  1  %  per  annum.  During  the  third  year  it 
was  destroyed  by  fire.  What  was  the  actual  loss  to  the  owner, 
no  allowance  being  made  for  interest? 

3.  A  man  purchased  goods  for  $  10,500  to  be  paid  in  3  equal 
installments,  without  interest ;  the  first  in  3  months,  the  sec- 
ond in  4  months,  the  third  in  8  months.  How  much  cash  will 
pay  the  debt,  money  being  worth  7  %  ? 

4.  The  surface  of  a  sphere  is  the  same  as  that  of  a  cube,  the 
edge  of  which  is  12  inches.     Find  the  volume  of  each. 

5.  Subtract  10^  from  15 J,  divide  the  remainder  by  |,  add 
.625  to  the  quotient,  multiply  this  sum  by  16J,  and  add  66^^ 
to  the  product. 

6.  A  square  field  contains  10  acres.  What  will  it  cost  to 
fence  it  at  $  1.25  per  rod  ? 

7.  The  longitude  of  Cincinnati  is  84  degrees  26  minutes  W., 
and  that  of  San  Francisco  122  degrees  26  minutes  15  seconds 
W.  When  it  is  noon  at  Cincinnati,  what  time  is  it  in  San 
Francisco  ? 

8.  How  many  pencils  7  inches  long  can  be  made  from  a 
block  of  red  cedar  7  inches  by  lOJ  inches  by  2J  inches,  if  the 
block  is  sawed  into  strips  3^  inches  wide  and  ^  inches  thick, 
each  strip  making  the  halves  of  6  pencils  ? 

9.  A  man  bought  a  horse  for  $  72,  and  sold  it  for  25  %  more 
than  cost,  and  10  %  less  than  he  asked  for  it.  What  did  he 
ask  for  it  ? 


146  MATHEMATICAL  WRINKLES 

10.  A  person  purchased  two  lots  of  land  for  $200  each,  and 
sold  one  at  40  %  more  than  cost,  and  the  other  at  20  %  less 
than  cost,  and  took  a  promissory  note  for  the  amount  of  the 
proceeds  of  the  sale,  bearing  8  %  interest  for  2  years  com- 
pounded annually.  At  maturity  he  collected  the  note.  What 
per  cent  of  profit  was  the  amount  of  the  note  on  the  original 
sum  invested  in  the  lots. 

State  Examination.  —  Oklahoma 

1.  How  do  you  teach  the  carrying  of  tens  in  addition  ? 

2.  Illustrate  by  a  drawing  of  a  dial  plate  that  the  time  past 
noon  plus  the  time  to  midnight  equals  12  hours. 

3.  Explain   the   process  of   multiplying   a   fraction   by   a 
fraction. 

4.  Explain  the  placing  of  the  decimal  point  in  multiplica- 
tion and  division  of  decimals. 

5.  Present  your  method  of  teaching  interest. 

6.  Factor  1225,  1448,  2356. 

7.  Illustrate  three  ways  of  finding  the  G.  C.  D. 

8.  Find  the  annual  interest  on  $  500  for  5  years  5  months 
and  5  days  at  6  % . 

9.  The  list  price  of  goods  is  $90.     I  buy  for  20  and  10  off. 
Find  cost  to  me. 

10.  The  diagonal  of  a  square  field  is  75  rods.  What  would 
be  the  diagonal  of  another  square  field  whose  area  is  four  times 
as  great  ?     Illustrate. 

11.  At  66  cents  a  bushel,  what  is  the  value  of  the  wheat 
which  fills  a  bin  6  feet  long  and  5  feet  square  at  the  ends  ? 

12.  The  boundaries  of  a  square  and  circle  are  each  40  feet. 
Which  has  the  greater  area  and  how  much  ? 


EXAMINATION  QUESTIONS  147 

For  Third  Grade  Certificate.  —  Rhode  Island 

1.  Explain  the  fact  that  multiplying  the  numerator  of  a 
fraction  or  dividing  the  denominator  by  a  whole  number  in- 
creases the  value  of  the  fraction. 

2.  Simplify  the  fraction  ("if  +  5  of  ^^  h-  2^. 

3.  Coffee  bought  for  20  cents  per  pound  shrinks  8J%. 
For  how  much  per  pound  must  I  sell  it  to  gain  10  %? 

4.  Two  men  are  working  8  hours  and  10  hours  per  day  at 
the  same  daily  wages.  After  working  3  days,  each  works  1 
hour  per  day  more  for  3  days.  If  the  amount  paid  for  the 
whole  work  is  $  20.28,  what  should  each  receive  ? 

5.  Three  kinds  of  tea  costing  68  cents,  86  cents  and  96 
cents  a  pound  are  mixed  in  equal  quantities  and  sold  for  90 
cents  a  pound.     Find  the  gain  per  cent. 

6.  If  a  square  lot  contains  640  acres  of  land,  how  many 
rods  of  fence  will  be  required  to  inclose  it  ? 

7.  The  population  of  a  town  in  1890  was  12,298,  a  decrease 
otS^fo  of  the  census  of  1880 ;  in  1880  there  was  an  increase 
of  7^  %  of  the  census  of  1870.  What  was  the  population  in 
1870? 

8.  Telegraph  poles  are  usually  placed  88  yards  apart.  Show 
that  if  a  passenger  in  a  railway  train  counts  the  number  of 
poles  passed  in  3  minutes,  this  number  will  express  the  rate 
of  the  train  in  miles  per  hour. 

9.  A  gives  B  a  note  for  $  100,  payable  in  60  days.  If  B 
has  the  note  discounted  at  a  bank  at  6  %  2  weeks  afterward, 
how  much  money  will  he  receive  ? 

10.  (a)  If  the  price  of  land  is  $  3000  per  acre,  what  would 
a  lot  60  feet  by  100  feet  cost  ?  (6)  What  would  be  the  cost  of 
a  similar  lot  50  feet  long  at  double  the  price  ? 


148  MATHEMATICAL   WHmKLES 

JuxiOR  Matriculation. — Ontario 

1.  Express  as  a  decimal : 

/2^         8.8   \  .  /1.74       ^    \_5 
(,  6.3        0.0625J  •  Vl2.2i  V      6' 

2.  Use  contracted  methods  to  find : 

(a)  1250  (1.05)^,  correct  to  two  decimal  places ; 
(&)  1  -=-  0.4342945,  correct  to  four  decimal  places. 

3.  How  much  money  deposited  in  a  bank  will  amount  to 
$  1500  in  1  year,  the  bank  paying  3  %  per  annum,  compounded 
quarterly  ? 

4.  A  man  has  a  choice  of  insuring  his  house  for  |  of  its 
value  at  li  %,  or  for  ^  of  its  value  at  1\%.  By  what  per  cent 
of  the  value  of  the  house  is  one  premium  greater  than  the  other  ? 

5.  What  is  the  value  of  the  goods  handled  in  each  of  the 
following  cases : 

(a)  An  agent  receives  $2450  to  invest  in  goods  after  re- 
taining his  commission  of  2^%? 

(6)  An  agent  remits  to  his  firm  $2450,  the  proceeds  of  a 
sale  for  which  he  retains  his  commission  of  2^  %  ? 

6.  A  man  has  an  annual  income  of  $1785  from  an  invest- 
ment in  10|^  %  stock  which  is  quoted  at  137.  What  would  his 
income  be  if  he  had  his  money  out  at  7  %  interest  ? 

7.  What  must  a  Canadian  company  pay  for  a  draft  to  can- 
cel a  debt  of  £  2430  in  London,  Eng.,  exchange  being  quoted 
at  8^? 

8.  The  base  of  a  prism  of  height  125  inches  is  a  parallelo- 
gram with  a  diagonal  104  inches  and  two  sides  45  inches  and 
85  inches.     Find  the  volume. 

9.  Find  (a)  the  total  surface,  (5)  the  volume,  of  a  block  of 
wood  18  inches  square  and  3  inches  thick,  with  a  circular  hole 
of  14  inches  diameter  through  its  center. 


EXAMINATION  QCJESTIONS  149 

State  Examination.  —  North  Dakota 

1.  Define  commission,  interest,  exchange,  annuity. 

2.  A  boy  who  bought  20%  as  many  marbles  as  he  had, 
found  that  he  then  had  60.     How  many  had  he  at  first  ? 

3.  According  to  the  metric  system  what  is  the  unit  of 
capacity,  of  weight,  of  surface  measurements  ? 

4.  What  must  be  the  length  of  a  plot  of  ground,  if  the 
breadth  is  18|  feet,  that  its  area  may  contain  56  square  yards  ? 

5.  What  must  be  the  price  paid  for  5  %  stock  so  that  it 
may  yield  the  same  rate  of  income  as  4|  %  stock  at  96  ? 

6.  A  merchaht  sold  a  coat  for  $15.40  and  gained  20%. 
How  much  would  he  have  gained  if  he  had  sold  it  for  $  16.50  ? 

7.  What  is  the  depth  of  a  cubical  cistern  which  contains 
2744  cubic  feet  ?  What  will  it  cost  to  plaster  the  sides  and 
bottom  at  $  .35  per  square  yard  ? 

8.  A  village  must  raise  $  8795  by  taxation.  The  assessed 
valuation  is  $989,387,  and  there  are  670  persons  subject  to  a 
poll  tax  of  $1  each.  A's  property  is  assessed  at  $10,000 
and  he  is  a  resident  of  the  village.  What  amount  will  he  pay 
in  taxes  ? 

9.  A  bridge  is  6  rods  long  and  18  feet  wide.  What  is  the 
cost  of  flooring  this  bridge  with  3-inch  plank  at  $  22.50  per  M.? 

10.   A  has  I  more  money  than  B,  and  together  they  have 
$510.     How  much  has  each  ?     Give  work  in  full. 

For  Teachers'  Certificate.  —  Iowa 

1.  On  a  map  constructed  on  a  scale  of  y^TrJ^^n^  the  dis- 
tance from  Detroit  to  Chicago  is  11.29  inches.  How  many 
miles  between  these  cities  ? 

2.  W^hat  principal  will  yield  $62,50  interest  in  1  year 
3  months  at  4  %  ? 


150  MATHEMATICAL   WRINKLES 

3.  Define :  concrete  number,  interest,  gram,  date  line,  cord 
foot. 

4.  (a)  Divide  SJ  -  |  X  A  by  21i  +  J,  +  4^  x  5. 

(6)  What  decimal  part  of  a  bushel  is  2  pecks  4  quarts? 

5.  What  is  the  area  of  the  circle  inscribed  in  a  square 
whose  area  is  196  square  inches  ?  Of  the  square  inscribed  in 
this  circle  ? 

6.  A  collector  has  a  $500  note  placed  in  his  hands  with 
power  to  compromise;  he  accepts  75  cents  on  a  dollar  and 
charges  5  %  of  the  sum  collected,  and  25  cents  for  a  draft. 
What  are  the  net  proceeds  ? 

7.  What  is  the  difference  between  local  time  and  standard 
time  at  Chicago,  the  longitude  of  Chicago  being  87  degrees 
36  minutes  and  42  seconds  west  ? 

8.  Which  is  the  better  discount,  10%,  12%,  5%,  or  15%, 
6  %,  6  %  ?  What  three  equal  rates  of  discount  are  equivalent 
to  the  latter  ? 

9.  A  cubic  foot  of  water  weighs  1000  ounces,  and  in  freez- 
ing expands  -^  of  itself  in  length,  breadth,  and  thickness. 
Eind  the  weight  of  a  cubic  foot  of  ice. 

10.  When  a  Boston  draft  for  $  35,000  can  be  bought  in  New 
Orleans  for  $34,930,  is  exchange  at  a  premium,  at  par,  or  at 
a  discount  ?     What  is  the  rate  ? 

Teachers'  State  Examination.  —  Iowa 

1.  Define :  composite  number,  concrete  number,  least  com- 
mon multiple  of  two  or  more  numbers,  rectangle,  trapezoid, 
common  fraction,  decimal  fraction. 

2.  (a)  Express  in  Roman  notation :  723,  1909,  1776,  2499, 
31,749. 

(b)  Express  in  words :  .0276,  100.001,  101,  .00047. 


EXAMINATION  QUESTIONS  151 

3.  Reduce  44  rods  5  feet  6  inches  to  the  decimal  part  of  a 
mile. 

4.  How  many  yards  of  carpet  27  inches  wide  will  be  needed 
to  carpet  a  room  13  feet  by  17  feet  if  the  waste  in  matching  is 
6  inches  on  a  strip  ? 

5.  If  goods  are  bought  at  20  and  10  %  off  and  sold  at  list 
price,  what  per  cent  of  profit  is  made  ? 

6.  A  note  for  $  580  dated  March  16,  1909,  and  due  in  one 
year  at  6  %  interest,  was  discounted  at  a  bank  3  months  later 
at  8  % .     Find  the  proceeds. 

7.  A  water  tank  is  16  feet  long,  4  feet  wide,  and  2^  feet 
high.     How  many  barrels  will  it  hold  ?     How  many  bushels  ? 

8.  Find  the  number  of  acres  within  a  circular  race  track 
whose  circumference  is  |  of  a  mile. 

9.  A  tax  of  $52,000  is  to  be  raised  in  a  city  whose  assessed 
property  valuation  is  $1,830,000.  Find  the  tax  rate.  If  A's 
property  is  assessed  at  $16,000,  how  much  does  he  pay  for 
his  taxes  ? 

10.  A  factory  valued  at  $50,000  was  insured  for  J  of  its 
value  at  J%  premium.     Find  the  annual  premium. 

11.  Find  the  diagonal  of  a  field  that  is  a  half  mile  long  and 
contains  120  acres.  How  many  rods  of  fence  will  be  needed 
to  inclose  this  field  ? 

For  State  Certificate.  —  South  Carolina 

1.  Divide  7.601825  by  347.512,  multiply  quotient  by  .05, 
to  the  product  add  3.45,  and  from  sum  subtract  2.115. 

2.  Simplify  (3i  +  4i-5i  x  f)^(3J). 

3.  Find  the  weight  in  tons  of  the  water  in  a  dock  24  feet 
deep  and  covering  j\  of  an  acre,  given  that  a  cubic  foot  of 
water  weighs  62^  pounds. 


152  MATHEMATICAL   WRINKLES 

4.  Find  the  simple  interest  on  $2000  for  2  years  9  months 
18  days  at  7%. 

5.  How  many  men  are  required  to  cultivate  a  field  of  7|- 
acres  in  5^  days  of  10  hours  each  ?  Given  that  each  man  com- 
pletes 77  square  yards  in  9  hours. 

6.  On  a  map  made  on  a  scale  of  6  inches  to  a  mile,  a  rect- 
angular field  is  represented  by  a  space  1  inch  long  and  J  inch 
broad.     How  many  acres  are  there  in  the  field  ? 

7.  At  what  rate  per  cent  will  $2250  amount  to  $2565  in 
4  years  at  simple  interest? 

8.  If  the  wholesale  dealer  makes  a  profit  of  25  %  and  the 
retail  dealer  a  profit  of  40%,  what  is  the  cost  of  an  article 
which  is  sold  at  retail  for  $  18  ? 

9.  What  fraction  of  39  gallons  is  3  bushels  and  3  pints  ? 
If  a  gallon  contains  231  cubic  inches  and  a  bushel  contains 
2150.4  cubic  inches,  answer  as  a  common  fraction  in  its  lowest 
terms. 

State  Examination.  —  Virginia 

1.  (a)  A  fruit  dealer  bought  oranges  at  the  rate  of  40  for 
$1,  and  sold  them  at  50  cents  per  dozen.  Find  gain  per 
cent.  (6)  He  also  bought  apples  at  the  rate  of  5  for  2  cents, 
and  sold  them  at  8  cents  per  dozen.  How  many  must  he  buy 
and  sell  in  order  to  gain  $2?     (Analyze  in  full.) 

2.  The  steeple  of  a  certain  church  is  a  pyramid  28  feet  in 
slant  height,  and  stands  upon  a  base  14  feet  square.  Find 
cost  of  painting  it  at  10  cents  per  square  yard. 

3.  A  certain  city  bought  two  horses  for  the  fire  department, 
but  finding  them  unfit  for  the  work,  sold  them  for  $  300  each : 
thus  gaining  20  per  cent  on  one,  and  losing  20  per  cent  on  the 
other.  Did  the  city  gain  or  lose,  and  how  much?  (Show 
work.) 


EXAMINATION  QUESTIONS  153 

4.  A  rectangular  lot  contains  one  acre  and  has  a  street 
frontage  of  120  feet.  How  deep  is  the  lot  and  how  many 
yards  of  fence  are  required  to  inclose  it  ? 

5.  (a)  What  is  16^%  of  900?  (b)  98  is  what  per  cent  of 
2450?  (c)  128  is  32%  of  what  number?  (d)  1350  is  25% 
more  than  what  number?  (e)  765  is  10%  less  than  what 
number  ? 

6.  Find  the  interest  and  n\aturity  value  of  a  note  of  $600 
for  3  years  3  months  24  days  at  6  %. 

7.  (a)  Write  a  negotiable  promissory  note,  using  the  above 
data,  (b)  Make  out  a  bill  containing  four  items  of  merchan- 
dise, and  acknowledge  payment. 

8.  A  man  sold  his  farm  and  invested  the  money  at  6% 
interest.  In  one  year  he  spent  ^  of  his  income  traveling,  ^ 
for  a  library,  and  saved  $  100.  Kequired,  selling  price  of  farm. 
(Analyze  in  full.) 

9.  A  wagon  loaded  with  hay  weighed  43  hundredweight 
and  GS  pounds.  The  wagon  was  afterwards  found  to  weigh 
9  hundredweight  and  98  pounds.  Required,  value  of  hay  at 
$  10  per  ton. 

10.   What  is  the  net  amount  of  a  bill  of  S  800,  after  allow- 
ing successive  discounts  of  25  %,  10  %,  and  5  %  ? 

Second  Class  Professional.  —  Ontario 

1.  Write  an  article  on  Arithmetic  in  Public  Schools,  under 
the  following  headings : 

(a)  Purpose  of  teaching  Arithmetic ; 
(6)  Correlation  with  other  subjects ; 
(c)  Place  and  value  of  Oral  Arithmetic. 

2.  Outline  a  lesson  plan  for  teaching  "  8 "  (Numbers  1-7 
are  supposed  to  be  known).  What  facts  would  you  teach 
before  proceeding  to  "  9  "  ? 


154  MATHEMATICAL   WRINKLES 

3.  Assuming  that  your  class  know  how  to  multiply  by  a 
one  digit  number,  show  how  you  would  teach  the  multiplication 
of  234  by  23. 

4.  How  would  you  make  clear  to  a  class  the  principles  in- 
volved in  the  ordinary  method  of  finding  the  G.  C.  M.  of  such 
numbers  as  2449  and  2573  ? 

5.  Mention  the  topics  of  all  the  previous  lessons  in  frac- 
tions which  you  would  require  to  teach  as  a  preparation  for  a 
lesson  on  the  multiplication  of  -f-  by  f.  Outline  your  plan  for 
this  lesson. 

6.  Solve,  as  you  would  for  your  pupils,  the  following : 

(a)  Find  the  square  root  of  272-^5-. 

(6)  A  man  has  $  6250  6  %  stock  and  sells  it  at  80.  With 
the  proceeds  he  buys  a  house  on  which  he  pays  insurance  at 
J%  per  annum  on  4-  of  its  value,  and  taxes  at  20  mills  on 
the  dollar  on  $4500  assessment,  and  in  addition  a  water 
rate  of  $11  per  annum.  If  he  rents  the  house,  what  monthly 
rent  should  he  charge  that  his  annual  income  may  be  the  same 
as  that  derived  from  the  stock  ? 

(c)  An  agent  sells  1000  barrels  of  flour  at  $5.50  a  barrel, 
and  charges  2^%  commission;  expenses  for  freight,  etc.,  are 
$500.  With  the  net  proceeds  he  buys  sugar  at  6\  cents  a 
pound,  charging  2i  %  commission.  How  much  sugar  does  he 
buy? 

(d)  A  ditch  has  to  be  made  360  feet  long,  8  feet  wide  at 
the  top,  and  2  feet  wide  at  the  bottom ;  the  angle  of  the  slope 
at  each  side  being  45°.  Find  the  number  of  cubic  yards  to  be 
excavated. 

For  State  Certificate. — North  Dakota 

1.  Define  Arithmetic,  numeration,  compound  number,  inter- 
est, per  cent. 


EXAMINATION  QUESTIONS  155 

2.  A  farmer  sold  a  horse  for  $  80  and  lost  20  %  of  its  cost. 
He  then  bought  a  horse  for  $80  and  afterward  sold  it  at  a 
gain  of  20%.  How  much  did  he  gain  or  lose  on  the  two 
transactions  ? 

3.  Multiply  the  sum  of  }  and  ^  by  their  product  and 
reduce  the  result  to  a  decimal. 

4.  Explain  the  difference  between  a  common  and  a  decimal 
fraction. 

5.  The  product  of  three  numbers  is  420,  and  two  of  the 
numbers  are  5  and  7.     Find  the  third  number. 

6.  Find  the  value  of  (J  +  ^)  x  -^  +  (^^  -f  f  X  3). 

7.  How  many  acres  in  a  strip  of  land  80  rods  long  and  14 
rods  wide? 

8.  C  and  D  together  own  921  acres  of  land,  of  which  C 
owns  420  acres.  C's  land  equals  what  fractional  part  of  D's  ? 
D's  land  is  what  per  cent  of  the  whole  ? 

9.  What  will  be  the  cost  of  the  wood  that  can  be  piled  in 
a  shed  20  feet  long,  10  feet  wide,  and  8  feet  high,  at  $4.75  per 
cord? 

10.    The  longitude  of  Constantinople  is  28°  59'  E.     When  it 
is  noon  in  Greenwich,  what  is  the  time  in  Constantinople  ? 

Teachers'  Certificate.  —  Arkansas 

1.  A  man  bought  a  horse  and  paid  J  of  the  price  in  cash. 
One  year  later  he  paid  J  of  what  remained,  and  the  two  pay- 
ments amounted  to  S 1530.     What  was  the  price  of  the  horse  ? 

2.  A  having  lost  25  %  of  his  capital  is  worth  as  much  as 
B,  who  has  just  gained  15%  on  his  capital;  B's  capital  was 
originally  $5000.     What  was  A's  capital  ? 

3.  A  square  field  contains  131  acres  65  square  rods. 
What  will  it  cost  to  fence  it  at  62i  cents  a  rod  ? 


156  MATHEMATICAL   WRINKLES 

4.  The  width  of  a  river  is  100  yards  and  it  averages  5  feet 
in  depth.  Find  the  number  of  cubic  feet  of  water  which  flows 
past  a  given  point  in  one  minute  if  the  average  rate  of  the 
stream  is  2|-  miles  per  hour. 

5.  A  man  bought  oranges  at  the  rate  of  3  for  2  cents,  and 
an  equal  number  at  the  rate  of  4  for  3  cents.  He  sold  them 
at  the  rate  of  2  for  5  cents  and  gained  $4.30.  How  many- 
oranges  did  he  buy  ? 

6.  A  man  divided  $  500  among  his  three  sons,  so  that  the 
second  had  -f  as  much  as  the  first,  and  the  third  f  as  much  as 
the  second.     How  much  did  each  receive  ? 

7.  A  clock  is  set  at  12  o'clock  Monday  noon,  and  on 
Tuesday  morning  at  9  o'clock  it  had  lost  3  minutes.  What  will 
be  the  correct  time  when  it  strikes  3  o'clock  the  next  Friday 
afternoon  ? 

8.  Find  the  interest  on  $9430  for  2  years  5  months  7  days 
at  5%,  using  the  method  which  you  believe  best  adapted  for 
class  use  in  teaching  interest. 

9.  The  catalogue  price  of  a  book  is  $  3.  If  I  buy  it  at  a 
discount  of  40%  and  sell  it  at  20%  below  catalogue  price, 
what  is  my  gain  per  cent  ? 

10.    A  and  B  together  have  $  153 ;  f  of  A's  money  equals  f 
of  B's.     How  much  has  each?     (Write  full  analysis.) 

Ontario  Examination  Questions.  —  University 
Matriculation 

1.  From  1870  to  1880,  the  population  of  a  town  increased 
30  %  ;  from  1880  to  1890  it  decreased  30  %.  The  population 
in  1870  exceeded  tiiat  in  1890  by  2781.  Find  the  population 
in  1880. 

2.  (a)  A  man  borrows  $  12,000  for  a  year  at  8  %  and  loans 
it  at  2  %  per  quarter  year,  compounding  interest  at  the  end 


EXAMIKATION  QUESTIONS  157 

of  each  quarter.    How  much  money  will  he  have  made  at  the 
end  of  the  year  ? 

(b)  A  borrows  from  B  a  sum  of  money  and  agrees  to  pay 
him  by  three  annual  payments  of  $200  each.  If  money  is 
worth  5  %  per  annum,  compound  interest,  find  the  sum  bor- 
rowed. 

3.  A  commission  merchant  received  500  barrels  of  flour, 
which  he  sold  at  S5  a  barrel,  charging  2%  commission;  he 
was  instructed  to  invest  the  net  proceeds,  deducting  a  purchase 
commission  of  2  %,  in  tea.  Find  the  value  of  the  tea  bought, 
and  the  total  commission. 

4.  A  man  holds  $  15,600  stock  worth  60 ;  to  transfer  to  4  % 
stock  at  78  will  increase  his  annual  income  $  12 ;  he  effects 
the  transfer,  but  not  until  each  stock  has  increased  2  in  price. 
Find  the  increase  of  his  income. 

5.  A  merchant  marks  his  goods  at  an  advance  of  25  %  on 
cost.  After  selling  \  of  the  goods,  he  finds  that  some  of  the 
goods  in  hand  are  damaged  so  as  to  be  worthless;  he  marks 
the  salable  goods  at  an  advance  of  10  %  on  the  marked  price 
and  finds  in  the  end  that  he  has  made  20  %  on  cost.  What 
part  of  the  goods  was  damaged  ? 

6.  A  grocer,  by  selling  12  pounds  of  sugar  for  a  certain 
sum,  gained  20  %.  If  sugar  advances  10  %  in  the  wholesale 
market,  what  per  cent  will  the  grocer  now  gain  by  selling  10 
pounds  for  the  same  sum  ? 

7.  A  note  made  June  1,  at  3  months,  was  discounted  imme- 
diately at  8  %  per  annum,  and  produced  $  357.40.  What  was 
the  face  of  the  note  ? 

8.  ^Vhat  rate  per  cent  per  annum,  compounded  half-yearly, 
is  equivalent  to  6  %  per  annum,  compounded  yearly  ? 

9.  Two  candles  are  of  equal  length.  The  one  is  consumed 
uniformly  in  4  hours,  and  the  other  in  6  hours.    If  the  candles 


158  MATHEMATICAL   WRIKKLES 

are  lighted  at  the  same  time,  when  will  one  be  three  times 
as  long  as  the  other  ? 

10.  Calculate  the  number  of  acres  in  the  surface  of  the 
earth,  considering  the  earth  a  sphere  of  8000  miles  diameter. 

State  Examixation.  —  Ohio 

1.  I  have  three  pitchers  holding  respectively  IJ,  2},  and  S^ 
pints.  How  many  times  can  I  fill  each  from  the  smallest  keg 
that  will  hold  enough  to  fill  each  pitcher  an  exact  number  of 
times  ? 

2.  Bought  20  yards  cloth,  li  yards  wide,  at  $  2  per  yard. 
The  cloth  shrunk  20  %  in  length,  and  25  %  in  width.  At  what 
price  per  yard  must  I  now  sell  the  cloth  so  as  to  gain  20  %  ? 

3.  Bought  6  %  railroad  stock  at  109i-,  brokerage  i-  %.  AVhat 
must  the  same  stock  bring  6  years  later  to  pay  me  8  %  in- 
terest ? 

4.  A  and  B  form  a  partnership.  A  contributes  $  7000, 
and  is  to  have  |  of  the  profits ;  B  contributes  $  8000,  and  is 
to  have  ^  of  the  profits ;  each  partner  is  to  receive  or  pay 
interest  at  6  %  per  annum  for  any  excess  or  deficit  in  his  share 
of  capital.  At  the  end  of  the  first  year  the  profits  are  $  1800. 
Required  worth  of  each  share. 

5.  How  many  shares  of  stock  at  40  %  must  A  buy,  who  has 
bought  120  shares  at  74  %,  150  shares  at  68  %,  and  130  shares 
at  54  %,  so  that  he  may  sell  the  whole  at  60  %,  and  gain  20  %  ? 

6.  A  laborer  agreed  to  build  a  fence  on  the  following  condi- 
tions :  for  the  first  rod  he  was  to  have  6  cents,  with  an  increase 
of  4  cents  on  each  successive  rod;  the  last  rod  came  to  226 
cents.     How  many  rods  did  he  build  ? 

7.  A  wins  9  games  of  chess  of  15  when  playing  against  B, 
and  16  out  of  25  when  playing  against  C.  At  that  rate,  how 
many  games  out  of  118  should  C  win  when  playing  against  B  ? 


EXAMINATION   QUESTIONS  159 

8.  B  agreed  to  work  40  days  at  S  2  per  day,  and  board ;  but 
he  agreed  to  pay  $  1  a  day  for  board  each  day  that  he  was  idle. 
How  many  days  was  he  idle,  if  he  received  $  44  for  his  work 
during  the  40  days  ? 

Quarterly  Examination.  —  Gunter  Bible  College 

1.  Define  insurance,  arithmetical  progression,  geometrical 
progression,  and  arithmetical  complement. 

2.  What  is  the  distance  passed  through  by  a  ball  before  it 
comes  to  rest,  if  it  falls  from  a  height  of  40  feet  and  rebounds 
half  the  distance  at  each  fall  ? 

3.  A  merchant  adds  33^  %  to  the  cost  price  of  his  goods, 
and  gives  his  customers  a  discount  of  10  %.  What  profit  does 
he  make? 

4.  What  is  the  difference  between  the  simple  and  compound 
interest  on  $750  for  2  years  7  months,  at  5  %  ? 

6.  If  the  duty  on  linen  collars  and  cuffs  is  40  cents  per 
dozen  and  20  %,  what  is  the  duty  on  10  dozen  collars  at  75 
cents  a  dozen,  and  10  pairs  of  cuffs  at  25  cents  a  pair? 

6.  The  capital  stock  of  a  company  is  $1,000,000,  J  of  which 
is  preferred,  entitled  to  a  7  %  dividend,  and  the  rest  common. 
If  $47,500  is  distributed  in  dividends,  what  rate  of  dividend 
is  paid  on  the  common  stock? 

7.  Find  the  bank  discount  and  proceeds  of  a  90-day  note 
for  $  1500  at  6  %  interest,  dated  Aug.  10,  and  discounted 
Sept.  1,  at  7  %. 

8.  On  Jan.  1,  1908,  I  borrowed  $2000  at  10%  interest, 
paying  S300  every  3  months.  I  paid  the  debt  in  full  Jan. 
1,  1909.     AVhat  did  I  pay  by  the  United  States  rule? 

9.  Solve  No.  8,  by  the  Merchant's  rule.  Which  method  is 
'^^tte^  for  the  debtor  ?     Which  for  the  creditor  ? 


160  MATHEMATICAL   WRI]^KLES 

10.    Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.6990. 

(a)  Find  log  3^  x  51 

(6)  Find  the  number  of  digits  in  30^. 

Examination.     Arithmetic  A.  —  Gunter  Bible  College 

1.  Define  arithmetic,  bank  discount,  specific  gravity, 
involution,  commercial  discount,  and  ratio. 

2.  What  is  the  difference  in  area  between  a  square  whose 
diagonal  is  1  foot  and  a  circle  whose  diameter  is  1  foot? 

3.  In  a  lot  of  eggs  7  of  the  largest,  or  10  of  the  smallest, 
weigh  a  pound.  When  the  largest  are  worth  15  cents  a  dozen, 
what  are  the  smallest  worth  ? 

4.  My  wife's  age  plus  mine  equals  76  years,  and  |  of  her 
age  minus  2  years  equals  ^  of  my  age  plus  2  years.  Find  the 
age  of  each. 

5.  The  diameter  of  one  cannon  ball  is  2J  times  that  of 
another,  which  weighs  27  pounds.  What  is  the  larger  ball 
worth  at  1  cent  a  pound  ? 

6.  Bought  apples  at  SS  a  barrel.  Half  of  them  rotted. 
At  what  price  must  I  sell  the  remainder  in  order  to  gain  33^  % 
on  the  amount  bought  ? 

7.  Extract  the  cube  root  of  926,859,375. 

8.  A  uniform  rod  2  feet  long  weighs  1  pound.  What  weight 
must  be  hung  at  one  end  in  order  that  the  rod  may  balance  on 
a  point  3  inches  from  that  end  ? 

9.  In  any  year  show  that  the  same  days  of  the  month  in 
March  and  November  fall  on  the  same  day  of  the  week. 

10.  In  a  liter  jar  are  placed  1  kilogram  of  lead  and  1  kilo- 
gram of  copper.  What  volume  of  water  is  necessary  to  fill  the 
jar,  the  specific  gravity  of  lead  and  copper  being  respectively 
11.3  and  8.9? 


EXAMINATION  QUESTIONS  161 

11.  Bought  land  at  $60  an  acre.  How  much  must  I  ask  an 
acre  that  I  may  deduct  25  %  from  my  asking  price,  and  yet 
make  20  %  of  the  purchase  price  ? 

Advanced  Arithmetic.  —  Gunter  Bible  College 

1.  (a)  Define  arithmetical  complement,  bank  discount,  an 
equation,  specific  gravity,  tariff. 

(6)  Prove  (do  not  merely  illustrate)  that  to  divide  by  a  frac- 
tion one  may  multiply  by  the  divisor  inverted. 

2.  A  man  wishing  to  sell  a  horse  and  a  cow  asked  three 
times  as  much  for  the  horse  as  for  the  cow;  but  finding  no 
purchaser,  reduced  the  price  of  the  horse  20  %,  and  the  price 
of  the  cow  10  %,  and  sold  both  for  $  165.  How  much  did  he 
get  for  the  cow  ? 

3.  How  many  acres  are  in  a  square  the  diagonal  of  which 
is  20  rods  more  than  a  side  ? 

4.  (a)  Extract  the  sixth  root  of  1,073,741,824 

(P)  Simplify  pi4±l^±pq. 

5.  I  sold  a  book  at  a  loss  of  25  %.  Had  it  cost  me  $1 
more,  my  loss  would  have  been  40  %.     Find  its  cost. 

6.  (a)  Change  200332  in  the  quinary  scale  to  an  equiva- 
lent number  in  the  decimal  scale. 

(b)  Sum  to  infinity  the  series  I  +  Y  +  i+i+  "*• 

7.  If  100  grams  of  rock  salt  are  dissolved  in  1  liter  of  water 
without  increasing  its  volume,  what  will  be  the  specific  gravity 
of  the  solution  ? 

8.  (a)  If  a  ball  of  yarn  4  inches  in  diameter  makes  one 
pair  of  gloves,  how  many  similar  pairs  will  a  ball  8  inches  in 
diameter  make  ? 

(6)  What  must  be  paid  for  6  %  bonds  to  realize  an  income 
of  8  %  on  the  investment  ? 


162  MATHEMATICAL   WRINKLES 

9.  Find  the  difference  between  the  annual  interest  and 
compound  interest  of  $  6000  for  3  years  6  months  at  10  %. 

10.  An  article  cost  S  6.  At  what  price  must  it  be  marked  so 
that  the  marked  price  may  be  reduced  22  %  and  still  30  %  be 
gained  ? 

11.  At  what  two  times  between  3  and  4  o'clock  are  the  hour 
and  minute  hands  of  a  clock  equally  distant  from  12  ? 

For  State  Certificate.  —  New  Jersey 

Commercial  Arithmetic 

1.  Bought  of  Brown  &  Company  the  following  bill  of  lum- 
ber: 8750  feet  of  boards  at  $31,331  per  M.  feet;  5750 
shingles  at  $5.25  per  M. ;  2860  laths  at  $2.87i  per  M. ;  520 
joists,  20  feet  long,  16  inches  wide,  and  S^  inches  thick,  at  $15 
per  M.  feet.     Find  the  amount  of  the  bill. 

2.  Find  the  sale  price  of  a  Brussels  carpet  27  inches  wide 
at  $  1.60  per  yard  for  a  room  15  feet  long  and  13^  feet  wide 
if  the  strips  run  lengthwise. 

3.  Which  will  cost  the  more  and  how  much,  to  lay  a  brick 
sidewalk  260  feet  long  and  4i  feet  wide,  estimating  8  bricks 
for  each  square  foot  of  pavement  at  $  12  per  M.,  or  to  lay  a 
flagstone  walk  at  22  cents  per  square  foot  ? 

4.  How  much  will  it  cost  to  build  two  abutments  for  a 
bridge  each  18  feet  long  at  top  and  bottom,  12  feet  wide  at 
bottom  and  eight  (8)  feet  wide  at  top  and  11  feet  high  at  $  4.50 
a  perch  for  labor  and  stone  ? 

5.  Three  men  engaged  in  business.  A  furnished  $6000 
of  capital;  B  $9600,  and  C  $6400.  They  made  a  gain  of 
$4800  and  then  sold  out  the  business  for  $30,000.  What  was 
each  one's  share  of  gain  ? 

6.  What  must  I  pay  for  a  draft  on  Chicago  for  $475,  pay- 
able 30  days  after  date,  ^  %  premium,  interest  at  6  %  ? 


ANSWERS  AND  SOLUTIONS 

ARITHMETICAL  PROBLEMS 

1.  (a)  Let  f  =  distance  the  minute  hand  is  ahead  of  the 
hour  hand ;  V  =  distance  the  minute  hand  moves  while  the 
hour  hand  travels  f ;  ^'  =  distance  both  travel  =  120  spaces ; 
J  =  -jig.  of  120  spaces  =  4-j^  spaces ;  |^  =  2  times  4y\  spaces  = 
^A  spaces,  the  number  of  spaces  the  minute  hand  is  in  advance 
of  the  hour  hand. 

ih)  Let  f  =  distance  the  hour  hand  has  moved  past  3 
^  =  distance  the  minute  hand  moved  during  the  same  time ; 
Jiyt  =  15  minutes  +  9^  minutes  +  f ;  ^  =  24y\  minutes ;  ^  = 
•^  of  24^  minutes  =  |J^  minutes ;  ^  =  24  times  f ^  minutes 
=  26|ft^  minutes,  past  3. 

(c)  Since  the  hands  changed  places,  the  minute  hand  fell 
short  9^  minutes  of  going  2  hours.  Therefore  it  was  26y®^ 
minutes  past  3  when  I  first  looked,  and  120  minutes  —  9^ 
minutes  later  =  17^2^  minutes  past  5,  when  I  looked  the  second 
time.  Ana. 

2.  The  broken  part  of  the  tree,  resting  with  the  upper  end 
on  the  ground  and  the  other  end  attached  to  the  stump,  forms 
the  hypotenuse  of  a  right  triangle,  of  which  the  base  is  40  feet, 
and  the  altitude  is  the  stump  of  the  tree.  The  height  of  the 
tree  may  be  found  by  the  following  rule,  based  on  a  demonstra- 
tion in  Geometry :  From  the  square  of  the  height  subtract 
the  square  of  the  base,  and  divide  the  difference  by  twice  the 
height.  The  height  in  this  case  is  the  height  of  the  tree  and 
not  the  height  of  the  stump.    Therefore  (120^  -  40^)  -5-  (120  X  2) 

163 


164 


MATHEMATICAL   WRINKLES 


=  53^,  height  of  the  stump.      Then  120  feet  -  531  feet  =  66| 
feet,  Ans. 

3.  The  total  number  of  dollars  =  80  times  the  number  of 
acres ;  or  20  times  the  number  of  acres  =  the  number  of  dollars 
on  one  side  of  the  boundary.  One  dollar  is  1^  inches  in  dia- 
meter ;  hence  f  of  20  times,  or  30  times,  the  number  of  acres 
=  the  number  of  inches  on  one  side ;  f  times  the  number  of 
acres  =  the  number  of  feet  on  one  side.  Therefore  (f  times 
the  number  of  acres)^  -h  43,560,  or  5  times  the  square  of  the 
number  of  acres  -=-  34,848,  =  the  number  of  acres ;  5  times  the 
square  of  the  number  of  acres  =  34,848  times  the  number  of 
acres ;  or,  34,848  -?-  5  =  6969.6,  the  number  of  acres  in  the 
field,  Ans. 

4.  Let  ABOD  represent  the  rectangular  field.  Now  sup- 
pose four  such  fields  arranged  in  the  form  of  a  square  by  plac- 

H  A  B      ing  the  short  side  of  one  against  the 

long  side  of  another,  inclosing  the 
square  DEFG,  as  shown  in  figure. 

(J  Draw  the  diagonals  AC,  CK,  KI, 
and  lA.  It  may  be  readily  shown 
that  ACKI  is  a  square ;  and  since  a 
diagonal  is  100  rods,  the  area  of  the 
square  ACKI  =  10,000  square  rods. 
One  of  the  triangles,  as  ACB,  has 

'      an  area  of  15  acres,  or  2400  square 


/  ^ 

/         E 

\^ 

F  1 

K 


J 

rods.  Hence  the  combined  area  of  the  4  outer  triangles 
=  4  X  2400=  9600  square  rods.  Adding  this  result  to  the  area 
of  the  square  ACKI,  we  have  19,600  square  rods  =  the  area  of 
the  square  HBLJ.     Hence,  BL  =  Vl9,600  =  140  rods. 

Now  from  the  area  of  the  square  ACKI,  subtract  the  area  of 
the  four  inner  triangles,  and  we  have  the  area  of  the  square 
DGFEz^^OO  square  rods.  Hence  (?i^  =  V400  =  20  rods. 
Therefore,  BC  =  (140  -  20)  --  2  =  60  rods,  and  AB  =  60  +  20 
=  80  rods,  Ans. 


ANSWERS   AND   SOLUTIONS 


165 


6.  Let  A  and  O  be  the  points  the  candles  burn  to,  when  No. 
2  is  4  times  No.  1.  If  CD  be  used  as  a  unit  of  measure,  AB 
will  be  equal  to  4  such  units. 

Now,  if  the  candles  be  allowed 
to  burn  until  CD  is  consumed,  \  of 
a  unit  of  AB  will  burn,  leaving  3^ 
units.     Since  the  candles  have  been 
burning  4  hours,  the  3J  units  re- 
maining in  No.  2  will  be  consumed 
in  one  hour  if  they  continue  to  burn. 
Then  5  x  3  J  units  =  16  units,  the    handle  No.  i. 
number  of  units   in   each  candle.         ^o.  i  bums  in  4  hours. 
Then  since  16  units  of  No.  1  burn         ^^-  ^  ^""""^  ^"  '  ^^«"''«- 
in  4  hours,  16  units  of  No.  1,  or  the  part  consumed,  is  burned 
in  II  of  4  hours  =  3}  hours,  Arts. 

6.  The  distance  the  lizard  moves  is  the  hypotenuse  of  a 
right  triangle  whose  legs  are  200  feet  and  10  feet. 

^ns.  =  200.25  feet. 


n 

Candle  No.  2. 


$1 


$3.50 

\ 

2 

16 

1.50 

2 

2 

2 

.50 

2 

2 

10 

2 

80 

2 

16  calves. 
2  sheep. 


_82  lambs. 
100  number  of  head 
Another  answer  is  10,  20,  and  70. 

8.  The  daughter's  share  =  daughter's  share. 
The  wife's  share         =  2  daughter's  share. 
The  son's  share =  4  daughter's  share. 

D.'s  H-  W.'s  -f  S.'s  =  7  times  daughter's  share. 
*.  7  times  daughter's  share  =  the  estate. 

.*.  daughter's  share  =  |  of  the  estate, 

wife's  share  =  ^  of  the  estate, 

and  son's  share  =  ^  of  the  estate. 

9.  64. 


166  MATHEMATICAL   WRINKLES 

10.    The  distance  AB  is  the  hypotenuse  of  the  right  triangle 


ABC=  V(32)2  _^  (24)2  ^  40  ^^^^^ 


End     1^ 

Ceiling 

i 

j 

Side-wall 

1 
1 

\ 

End 

(' 

Floor 

11.  GC   %. 

12.  First,  let  us  find  the  volume  of  the  largest  ball  that 
j^  E  C     could  be  placed  in  the  given  cone  and  also 

the  amount  of  water  required  to  cover  it. 
Let  AC  in  the  diagram  represent  the 
diameter  of  the  mouth  of  the  glass,  BE  = 
4  inches,  the  altitude,  and  OZ>=the  ra- 
dius of  the  largest  marble  which  could  be 
covered  in  the  glass. 

Area  of  A  ABC  =  area  of  3  A  of  which 
OD  is  the  altitude.  Area  of  A  ABC  =  12. 
Hence  one  half  the  radius  of  the  largest  marble  =  12  h-  (5+5  -f 
6)  =  |. 

The  diameter  of  the  largest  ball  which  could  be  covered  in 
the  glass  =  3. 

=  12  7r. 


. •.  V  of  cone  ABC  =  -  iri^h  =  ^ 
3  3 


V  of  largest  marble  =  -  ird?  =  — . 
6  2 


3.2  TT ^  =  amount  of  water  it  takes  to  cover  the  largest 


ANSWERS   AND   SOLUTIONS  167 

marble.  Now,  the  water  which  would  cover  the  largest  marble 
is  to  the  water  covering  the  required  marble  as  the  largest 
marble  is  to  the  required  marble. 

.'.  a;  =  2.433  inches,  Ans. 

13.  The  discount  is  yj^  of  the  face  of  the  note.  The  interest 
is  10  %  of  the  proceeds. 

Hence,  10  %  of  the  proceeds  =  9  %  of  the  face. 
jIj^  of  the  proceeds  =  ^  of  yf^  =  -j-A^  of  the  face. 
{%%  of  the  proceeds  =  100  x  -j-J^^  =  ^^%%  of  the  face, 
jj jj  _  ^o^<^  —  ^Y^^  =  ^^  of  the  face,  which  is  the  discount 
for  the  required  time. 

.-.  the  time  is  ^  -i-  yj^  =  IJ  years  =  400  days. 

14.  Since  it  was  a  perfect  power,  the  right-hand  period 
must  have  been  25,  and  the  last  figure  of  the  root  must  have 
been  5.  Hence  the  last  trial  divisor  was  1225  -*-  5  =  245.  Then 
by  the  rule  for  extracting  the  square  root,  we  know  5  to  have 
been  annexed  to  24,  which  must  have  been  double  the  root 
already  found.  That  portion  of  the  root,  then,  must  have  been 
^  of  24  =  12.  The  entire  root  was  125.  Therefore  the  power 
was  125^=15,625,  Ans. 

15.  The  length  of  the  lawn  is  f  of  its  width,  and  if  J  of  it 
be  taken  off  by  a  line  parallel  to  the  end,  a  square  will  be  left, 
the  side  of  which  is  the  width  of  the  lawn.  The  area  of  the 
lawn  =  f  the  area  of  the  square.  If  the  dimensions  of  the  lawn 
be  increased  1  ft.,  its  area  will  be  equivalent  to  the  area  of  f 
of  the  square -ff  of  a  strip  1  foot  wide-|-|^  of  a  strip  1  foot 
wide  +  a  square  with  an  area  of  1  square  foot  =  651  square 
feet.  Area  of  f  of  the  square  -+-  f  of  a  strip  1  foot  wide  =  650 
square  feet.  Taking  J  of  this  quantity,  we  have  the  square 
+  1  of  a  strip  1  foot  wide  =  J  of  650  square  feet  =  62,400 
square  inches.     But  |  of  a  strip  1  foot  wide  =  a  strip  of  the 


168  MATHEMATICAL   WRINKLES 

same  length  |  of  a  foot  wide  =  two  strips  the  same  length  |-  of 
a  foot  wide,  or  10  inches  wide. 

Now,  if  we  place  these  two  strips  on  adjacent  sides  of  the 
square  and  also  a  square  containing  100  square  inches  at  the 
corner,  we  will  have  a  new  square  the  area  of  which  =  62,400 
square  inches  +  100  square  inches  =  62,500  square  inches.  A 
side  of  this  square  =  250  inches.  Therefore  the  width  of  the 
lawn  =  250  inches  —  10  inches  =  240  inches,  or  20  feet,  and 
the  length  =  30  ft.,  Ans. 

16.  Let  100  %  =  cost.  - 

180  %  =  marked  price. 

140  %  =  selling  price. 

.-.  i|-2.  =  1|-  =  length  in  yards. 

17.  The  area  of  a  triangle  whose  sides  are  13,  14,  and  15 
feet  may  be  found  by  the  rule :  "  Add  the  three  sides  together 
and  take  half  the  sum;  from  the  half  sum  subtract  each  side 
separately ;  multiply  the  half  sum  and  the  remainders  together 
and  extract  the  square  root  of  the  product.'' 

13  feet  -f  14  feet  + 15  feet  =  42  feet  =  sum  of  sides, 
i  of  42  feet  =  half  sum  of  sides. 
21  feet  -  13  feet  =  8  feet. 
21  feet  - 14  feet  =  7  feet. 
21  feet  - 15  feet  =  6  feet. 

7056  =  product  of  the  half  sum  and  the  three  remainders. 

•\/7056  =  84  square  feet,  area  of  triangle  with  sides  13,  14, 
and  15  feet. 

Area  of  given  triangle  is  24,276  square  feet. 

The  problem  now  becomes  merely  a  comparison  of  areas, 
the  larger  triangle  having  sides  in  the  same  proportion  as  the 
smaller.  Similar  surfaces  are  to  each  other  as  the  squares  of 
their  like  dimensions,  therefore,  84  :  24,276  :  :  13^ :  the  square 
of  the  corresponding  side.  Or,  V24,276  x  169  --  84  =  221, 
length  of  the  corresponding  side. 


ANSWERS   AND   SOLUTIONS  169 

Similarly  with  14  and  15  we  find  the  other  corresponding 
sides  =  238  and  255.  Aiis.  221,  238,  and  255. 

18.  Since  my  mistake  was  55  minutes,  the  hands  must  have 
been  5  minute  spaces  apart.  At  2  o'clock  they  were  10  spaces 
apart,  hence  the  minute  hand  had  gained  5  spaces.  It  gained 
55  spaces  in  1  hour,  hence  to  gain  5  spaces  requires  -jij-  of  an 
hour,  or  5^\  minutes.  Therefore,  it  was  5^  minutes  past 
2  o'clock,  Ans. 

19.  The  area  of  the  whole  slate  =  108  square  inches.  The 
area  of  the  frame  =  ^^  of  108  square  inches  =  27  square  inches. 
Now,  suppose  4  slates  so  placed  as  to  form  a  square  9  +  12,  or 
21  inches,  on  a  side.  The  whole  area  of  this  square  =  441 
square  inches.  441  square  inches  —  4  x  27  square  inches  = 
108  square  inches,  the  area  of  the  frames  of  the  four  slates  = 
338  square  inches,  the  area  of  a  square  formed  by  the  four 
slates  without  frames. 

V338  square  inches  =  18.242  inches,  a  side  of  the  square. 
Then  since  21  inches  includes  4  widths  of  the  frame,  21  inches 
—  18.242  inches  =  4  times  the  width  of  the  frame. 

Therefore  the  frame  is  .6895  inch  wide. 

20.  6  acres  +  72*  grdwths  keep  16  oxen  12  weeks,  or  1  ox 
for  192  weeks. 

3  acres  +  36  growths  keep  16  oxen  6  weeks,  or  1  ox  for  96 
weeks. 

Adding  the  above,  we  have, 

9  acres  -h  108  growths  keep  16  oxen  18  weeks,  or  1  ox  for 
288  weeks. 

9  acres +  81  growths  keeps  26  oxen  9  weeks,  or  1  ox  234 
weeks. 

Subtracting,  27  growths  keep  1  ox  288  weeks  —  234  weeks,  or 
54  weeks. 

.*.  1  growth  keeps  1  ox  2  weeks. 

.•.  IpO  growths  keep  1  ox  300  weeks. 

•  A  growth  is  the  weekly  growth  on  one  acre. 


170  MATHEMATICAL  WBINKLES 

Also,  72  growths  keep  1  ox  144  weeks. 

Then,  6  acres  keep  1  ox  192  weeks  —  144  weeks,  or  48  weeks. 
6  acres  keep  1  ox  48  weeks,  1  acre  keeps  1  ox  8  weeks. 
15  acres  keep  1  ox  120  weeks. 

.-.  15  acres  + 150  growths  keep  1  ox  120  weeks  +  300  weeks, 
or  420  weeks. 

Hence  the  number  of  oxen  required  is  420  ^  10  =  42,  Ans. 

21.  400. 

22.  Precedence  is  given  to  the  signs  X  and  ^  over  the  signs 
+  and  —  ;  hence  the  operations  of  multiplication  and  division 
should  always  be  performed  before  addition  and  subtraction. 
Ans.  =  8. 

23.  00  .  24.    0.  25.    2.236  minutes. 
26.   20  feet.                           27.   28.44+. 

28.  The  distance  from  the  extreme  point  of  the  given  ball 
to  the  corner  is  to  the  distance  of  the  nearest  point  of  the 
given  ball  from  the  corner,  as  the  diamfeter  of  the  given  ball  is 
to  the  diameter  of  the  required  ball. 

12  feet  =  an  edge  of  the  cube.  Then  V3  x  144  =  20.7846, 
the  distance  from  a  lower  to  the  opposite  upper  corner  of  the 
room.  20.7846  —  12  =  twice  the  distance  from  the  given  ball 
to  the  corner.  4.3923  =  the  distance  of  the  nearest  point  of  the 
given  ball  from  the  corner.  Then,  the  distance  from  the  extreme 
point  of  the  ball  to  the  corner  =  20.7846  -  4.3923  =  16.3923. 
.-.  16.3923  feet :  4.3923  feet : :  12  feet :  (3.215  feet),  Ayis. 

29.  Let  I  =  number  of  minutes  past  3  o'clock. 
40  —  |-  =  distance  the  minute  hand  is  from  8. 

y\  =  number  of  minute  spaces  the  hour  hand  is  from  3. 

15  4-  y\  =  distance  the  hour  hand  is  from  12.  But  since 
the  minute  hand  is  the  same  distance  from  8  that  the  hour 
hand  is  from  12,  then 

40—   24_-IK_|_     2 

.-.  I,  or  11  =  23yL  minute  past  3  o'clock,  Ans. 


ANSWERS   AND   SOLUTIONS  171 

30.  Since  an  edge  of  the  given  cube  differs  from  an  edge  of 
the  original  cube  by  2  inches,  the  difference  in  the  solidity  of 
the  cubes  will  be  the  solidity  of  7  blocks  2  inches  thick  —  a 
corner  cube,  3  narrow  blocks,  and  3  square  blocks.  The  con- 
tents of  these  7  solids  =  39,368  cubic  inches.  By  taking  away 
the  8  cubic  inches,  the  number  of  cubic  inches  in  the  corner 
cube,  there  remains  39,360  cubic  inches,  the  solidity  of  the 
3  narrow  blocks  and  3  square  blocks.  Then  1  square  block 
and  1  narrow  block  contain  13,120  cubic  inches. 

Now,  since  these  blocks  are  2  inches  in  thickness,  the  sum 
of  the  areas  of  1  face  in  each  of  the  2  =  6560  square  inches. 
That  is,  the  area  of  a  square  and  a  rectangle  2  inches  in  width 
=  6560  square  inches.  This  rectangle  is  equivalent  to  2  rect- 
angles of  equal  length  and  1  inch  wide.  Now,  if  we  place 
these  rectangles  on  adjacent  sides  of  the  square  and  also  add 
a  square  1  square  inch  in  area  to  complete  the  square,  we  will 
have  a  square  =  6561  square  inches.  A  side  of  this  square 
=  V6561  =  81  inches  =  an  edge  of  the  original  cube  after  the 
reduction,  increased  by  1  inch. 

.-.  an  edge  of  the  original  cube  =  82  inches,  Aiis. 

31.  The  required  number  is  the  remainder  left  after  sub- 
tracting the  largest  cube.  In  extracting  the  cube  root  of 
592,788  we  find  84  to  be  a  side  of  the  largest  cube,  and  84  to 
be  the  remainder.     .-.  84  is  the  required  number,  Ans, 

32.  80;  40. 

33.  In  1  hour  A  can  row  upstream  J  of  the  distance.  In 
1  hour  A  can  row  downstream  |  of  the  distance.  ^  —  i  =  J,  or 
twice  the  distance  the  stream  flows  in  1  hour.  Hence,  the 
stream  flows  -^  of  the  distance  in  1  hour.  ^  of  the  distance 
=  1  mile.     .*.  the  distance  =  12  miles,  Ans. 

34.  B's  share  =  i(90  4-  20)  =  22 ;  22  -  20  =  2,  the  loss. 

35.  The  required  number  is  the  remainder  left  after  sub- 
tracting the  largest  square.     In  extracting  the  square  root  of 


172  MATHEMATICAL   WRINKLES 

13,340  we  find  115  to  be  the  whole  mimber  of  the  root  and 
115  the  remainder.     .-.  115  is  the  required  number,  Ans. 

36.  v^number  =  10  Vnumber.  Raising  to  the  12th  power, 
(number)*  =  10^^  (number)^  Dividing  by  (number)^,  we  have 
the  number  =  10^^  ^  1,000,000,000,000. 

37.  -^.         38.    0.  39.    66|%. 

40.  This  problem  may  be  solved  by  Geometrical  Progres- 
sion. I  =  ar""'^.  I  =  Iff,  the  last  term,  a  =  1,  the  first  term, 
w  =  4,  the  number  of  terms. 

.•.|A|  =  r«,andr  =  |. 
.-.1-1  =  1,  or  121%. 

41.  $200;   $12.         42.    30.         43.    20%.         44.    $750. 
45.    $37,037.  46.  10: 40  o'clock.        47.   3iij  cords. 

48.  Let  100  %  =  cost  of  goods. 

180  %  =  marked  price  of  goods, 
iof  180%  =30%,  loss. 
.  180  %  -  30  %  =  150  %  =  selling  price. 
150  %  -  100  %  =  50  %,  gain,     Ans. 

49.  ^i  :  -^ : :  6  :  (5.738),  the  diameter  of  the  inside  sphere. 
6  inches  —  5.738  inches  =  .262  inch,  twice  the  thickness  of 
the  shell.     .*.  .131  inch  =  the  thickness  of  the  shell. 

50.  The  number  of  bushels  of  apples  =  |  of  20  bushels 
=  16  bushels. 

51.  Let  100  %  =  present  worth  of  sales. 

103  %  of  present  worth  of  sales  =  95  %  of  sales. 
1  %  of  present  worth  of  sales  =  ^%  %  of  sales. 
100  %  of  present  worth  of  sales  =  ^2^o%  %  of  sales. 
...  9232^4_  ^^  of  sales  =  1191%  of  cost  of  goods. 
1  %  of  sales  =  1.29if  J  %  of  cost  of  goods. 
100  %  of  sales  =  129if  ^  %  of  cost  of  goods. 
/.  29|f  J  %  =  the  per  cent  advance  of  the  cost. 


^^•90 


[    0    All  2 

190  a|i 


ANSWERS  AND  SOLUTIONS  173 

52.  J84,245,000  -  48,245,000  =  36,000,000. 

36,000,000  H-  36,000  =  1000,  the  divisor,     Ans. 

53.  1.754  inches;  2.246  inches;  4  inches. 

54.  3|  years.  56.    300  miles. 

56.    $300.  57.   60  days ;  40  days. 

58.  1600  -^  80  =  20,  the  difference  of  the  two  numbers.  The 
sum  -h  the  difference  =  twice  the  greater  number.  Hence, 
80  -f  20  =  100  =  twice  the  greater  number. 

.-.  50  =  the  greater  number,  and  30  =  the  smaller  number. 

59.  50%.  60.    15.  61.   3. 

10 
5 
10  gallons  of  water,    Ans. 

63.  Solve  by  means  of  Progression : 

Let  P  =■  principle ;  r=  rate  of  interest ;  n=  number  of  years ; 
A  =  amount  of  each  payment.     Then 

.^r.P(l-f  r)V 
(l  +  r)*-l 
Since  one  amount  is  paid  at  the  beginning  of  the  year,  the 
principal  less  that  amount  will  be  the  money  to  reckon  as  the 
new  principal  for  the  term  of  4  years. 

$1000-^=(P-^). 

(1+rr-l  (1+tV)*-1 

^tV($1000-^)(1.4641). 
.4641 
Clearing  of  fractions, 

$.4641  A=r^{$U64.1  -  1.4641  A). 
6.1051^=  $146.41. 

.♦.^=$239.81,   Ans. 

64.  3^%.  65.    $.67^. 


174  MATHEMATICAL  WRINKLES 

66.  The  true  discount  on  $lis  $!-($  1-^1.015)  =  $.014/^0^. 
The  bank  discount  on  $  1  is  $  .015. 

Then  $  .015  -  $  .014^\\\  =  mOj\\\,  the  difference. 
$  .90  ^  .000y2^  =  $  4060,  the  face  of  the  note,   Ans. 

67.  $50.  73.  11^  ounces. 

68.  8  days.  74.  3|-  years. 

69.  66|%.  75.  $160. 

70.  6  feet.  76.  25  dozen ;  92  cents. 

71.  168.298+ bushels.  77.  62.832  minutes. 

72.  8  pounds.  78.  5-^j  hours. 

79.  37  inches. 

80.  $76.52,  first;   $ 96.52,  second. 

81.  30  steps.  82    104  feet. 

83.  27^  minutes  after  5  o'clock. 

84.  lOif  minutes  past  2  o'clock. 

85.  A  pound  of  feathers. 

86.  600;  1200;  1800;  2400  yards. 

87.  360  acres.      88.   $20.  89.    1,000,000.      90.    $2. 

91.  Let  100  %=  the  marked  price. 
He  receives        100  %  -  10  %  =  90  % . 

Since  he  uses  a  yard  measure  .72  of  an  inch  too  short,  he 
gives  only  35 2V  inches  for  1  yard.  He  sells  35 Jg^  inches  for 
90  fc  of  the  marked  price.  Therefore  he  would  sell  36  inches 
for  91^  %  of  the  marked  price. 

.-.  100  %  —  91fi  %  =  8489  %,  the  required  discount. 

92.  69.36286+ pounds.  95.    $8.75. 

93.  32feet+.  96.    Book,  $1.10;  pen,  $.10. 

94.  8%.  97.    17%. 


ANSWERS  AND   SOLUTIONS  175 

98.  Each  new  day  begins  at  the  180th  meridian,  which  was 
crossed  in  the  Pacific  Ocean  before  reaching  Manila. 

99.  7  sheep.  101.   40.  103.   Friday. 
100.    f.                           102.    72.  104.    0. 

105.  3  P.M.  107.    G:40  p.m. 

106.  A,  S500;  B,  $700.  108.    B  paid  $92;  15%  gain. 

109.  The  greater  -f  the  less  =  582. 
The  greater  —  the  less  =  218. 

.-.  2  times  the  less  =  364, 
and  the  less  =  182. 
The  gi-eater  =  400. 

110.  2760.4288+  cubic  inches;  1152  square  inches. 

111.  A's,  $90;  B's,  $135;  G%  $180. 

112.  $20. 

113.  August  11  was  21  days  before  the  note  was  due. 
The  use  of  any  sum  of  money  for  21  days,  or  -j^^^  of  a  month, 
at  6  %  is  equal  to  -j-J^  of  it.  Then,  since  he  promised  to  pay 
such  a  sum  that  the  use  of  it  for  21  days  was  to  equal  the  use 
of  the  sum  unpaid  for  2  months,  y^  of  the  sum  unpaid  =  y^yVrr 
of  the  sum  paid.  Hence  the  sum  unpaid  =  -,^j^  of  the  sum 
paid  .-.  f^  of  the  sum  paid  +  ^%\  of  the  sum  paid  =  $  100. 
.-.  the  sum  paid  =  $  74.07. 

114.  $212.12.  118.  64  pounds. 

115.  $246.60.  119.  7  cents  to  A ;  1  cent  to  B 

116.  $50  gain.  120.  10. 

117.  43.817  pounds.  121.  45  feet. 

122.  30  of  first  quality ;  16  of  second  quality. 

123.  32  miles.  125.    810  revolutions. 

124.  $2;   SI.  126.    16  dozen. 


176  MATHEMATICAL    WRINKLES 

127.  A,  2.87  rods;  B,  4.72  rods;  C,  13.82  rods. 

128.  1,000,000.  134.  2. 

129.  Horse,  $110;  cow,  $10.  135.  20. 

130.  216  pounds.  136.  60. 

131.  $850.  137.  20%. 

132.  $4.  138.  20%. 

133.  They  are  the  same.  139.  First,  $250;  second, $200. 

140.  Husband's  age,  24  years ;  wife's  age,  20  years. 

141.  20  gallons  of  wine ;  30  gallons  of  water. 

142.  1300.       143.   4.       144.    $.80.       145.    $.75.        146.    8. 

147.  21-j9j-  minutes  past  4  o'clock. 

148.  lOif  minutes  past  2  o'clock. 

149.  27^  minutes  past  2  o'clock ;  3  o'clock. 

150.  43^^  minutes  past  2  o'clock. 

151.  .50.  152.   245.574. 

153.  Wife,  $8500;  son,  $12,750;  daughter,  $2125. 

154.  1  mile.  158.    9||%,  or  9.69+. 

155.  180.  159.    $42,949,672.95. 

156.  43,200.  160.    $4. 

157.  1,860,867.  161.    Midnight. 

162.   1  hour  and  20  minutes  is  lost  in  going  50  miles. 

.-.  80  minutes  is  lost  in  going  50  miles. 

.-.  1  minute  is  lost  in  going  |  mile. 

.-.  120  minutes  is  lost  in  going  75  miles. 

.-.  2  hours  is  lost  in  going  75  miles. 

But  2  hours  is  the  entire  time  lost. 

.*.  the  distance  traveled  after  the  breakdown  is  75  miles. 


ANSWERS  AND   SOLUTIONS  177 

Again,   the  train   at   its   original  speed   goes   as   far   in  3 
hours  as  it  went  in  5  hours  at  its  speed  after  the  breakdown, 
.-.  in  3  hours  at  the  original  speed  it  goes  75  miles. 
.-.  ill  1  hour  at  the  original  speed  it  goes  25  miles. 
.-.  the  length  of  the  line  is  75  miles  +25  miles  =  100  miles. 

163.   llj  cents.  166.   1^.  168.   300  feet. 

165.    2.  167.   4f  169.    2:1. 

170.  2  miles  340  feet. 

171.  132  and  140.       172.   20%.  173.    By  their  sum. 

174.  James's  speed  =  ^  of  my  speed. 
John's  speed  =  -J^J  of  James's  speed. 

.-.  James's  speed  =  1^  of  (^  of  my  speed). 
.-.  James's  speed  =  ^4|,  or  ^  of  my  speed. 
.-.  James's  speed  and  my  speed  are  in  the  ratio  of  456  to  500. 
.-.  in  running  500  yards  I  beat  James  500  yards  —  456  yards 
=  44  yards,  Ans. 

175.  First,  .759  inch;  second,  1.08+  inches ;,  third,  4.16  + 
inches. 

176.  First  Method.  1.  Any  remainder  which  exactly  di- 
vides the  previous  divisor  is  a  common  divisor  of  the  two 
given  quantities. 

2.  The  greatest  common  divisor  will  divide  each  remainder, 
and  cannot  be  greater  than  any  remainder. 

3.  Therefore,  any  remainder  which  exactly  divides  the  pre- 
vious divisor  is  the  greatest  common  divisor. 

Second  Method.  1.  Each  remainder  is  a  number  of  times  the 
greatest  common  divisor.  For  a  number  of  times  the  greatest 
common  divisor,  subtracted  from  another  number  of  times  the 
greatest  common  divisor,  leaves  a  number  of  times  the  greatest 
common  divisor. 

2.  A  remainder  cannot  exactly  divide  the  previous  divisor 
unless  such  remainder  is  once  the  greatest  common  divisor. 


178  MATHEMATICAL   WRINKLES 

3.  Hence,  the  remainder  which  exactly  divides  the  previous 
divisor,  is  once  the  greatest  common  divisor. 

177.  112  cubic  feet. 

178.  In  5  seconds  both  trains  travel  600  feet, 
.-.in  1  hour  both  trains  travel  Sl^  miles. 

In  15  seconds  the  faster  train  gains  600  feet. 

.-.  in  1  hour  the  faster  train  gains  27^  miles. 

Now,  we  have  the  sum  of  their  rates  =  81^^  miles  and  the 
difference  of  their  rates  =  27^  miles. 

.-.  rate  of  faster  +  rate  of  slower  =  81^^  miles,  and  rate  of 
faster  —  rate  of  slower  =  27^  miles. 

.-.  2  times  rate  of  faster  =  109 jij-  miles. 

.-.  rate  of  faster  ==  54^^  miles. 

Also,  2  times  rate  of  slower  =  54^^  miles. 


.-.  rate  of  slower  ==  27^^  miles. 

179.    1.118  times. 

185.   3042315V 

180.    SSte6. 

186.    3424^. 

181.    Senary. 

187.   139. 

182.   1,110,100,010  years. 

188.    124. 

183.   221446. 

189.    128. 

184.    10212^. 

190.    180. 

191.   658,548,918. 

192.   28  gallons  wine ;  42 

gallons  water. 

193.    lOf 

194.   Since  the  numbers  are  consecutive,  each  r 

the  cube  root  of  15,600 ;  in  other  words,  the  numbers  must  lie 
between  20  and  30.  Now,  15,600  is  divisible  by  25,  since  it 
ends  in  two  ciphers,  hence  25  may  be  one  of  the  numbers. 
By  trial,  we  find  that  624  would  be  the  product  of  the  other 
two,  which  themselves  must  end  in  4  and  6  to  give  a  product 
ending  in  4.  Ans.  24;  25;  26. 


ANSWERS  AND   SOLUTIONS  179 

195.  Such  a  number  must  lie  halfway  between  1042  and 
1236. 

.-.  1236  —  1042  =  194,  which  divided  by  2,  gives  97. 
.-.1042  +  97  =  1139,   Ans. 

196.  76,809,256,566. 

197.  49.  The  remainder  left  over  after  subtracting  the 
largest  cube  is  the  number. 

198.  At  4  miles  per  hour  =  1  mile  in  15  minutes,  and  5  miles 
per  hour  =  1  mile  in  12  minutes. 

.-.  in  going  1  mile  there  is  a  difference  of  3  minutes,  but  the 
actual  difference  is  10  minutes  +  5  minutes  =  15  minutes. 
.-.  15  minutes  -r-  3  minutes  =  5.  Ans.   5  miles. 

199.  i  of  small  glass  =  i  of  total,  and  since  the  large  glass 
is  J  of  both,  J  of  the  large  glass  =  J  of  total,  and  J  -f  J  =  ^ 
=  wine. 

.-.  |-J  =  water.  —  From  «  Arithmetical  Wrinkles." 

200.  When  the  ball  just  floats,  its  specific  gravity  is  1. 
Then  by  Allegation,  we  have 

\10|  9  I  35   I'  ^^  ^ 

^  of  t  ^ (12)»  =  Agf fi^  TT,  and  ^1[I|||^TTJ7) =5.52  inches, 
radius  of  ball,  and  12  —  5.62,  or  6.48,  inches  is  the  thickness  of 
the  shell.  —  From  "  The  School  Visitor." 

201.  14°  F.  =  -  10°  C.  and  270°  F.  =  132|°  C.  The  specific 
heat  of  ice  is  .505,  that  of  steam  is  .48,  latent  heat  of  fusion 
is  80,  and  that  of  evaporation  is  537 ;  then, 

100(10  X  .505  +  80  -f  100)  =  18,505  heat  units, 

required  to  melt  the  ice  and  raise  its  temperature  to  100°  C. 

There  are  80  x  32|  x  .48  =  1237^  heat  units  given  off  in 
reducing  the  steam  at  132|°  C.  to  steam  at  100°  C. 

There  are  (18,505-1237^)^537  =  32.16  pounds  of  steam 


180  MATHEMATICAL   WRINKLES 

at  100°  C.  to  be  condensed  to  water  at  100°  C.  The  result 
would  be  132.16  pounds  of  water  at  100°  C,  and  80-32.16 
=  47.84  pounds  of  steam  at  100°  C. 

—  From  "  The  School  Visitor." 

202.  If  the  average  for  the  entire  distance  were  30  miles  an 
hour,  50  X  30  or  1500  miles  would  be  run,  but  this  lacks  300 
miles  which  must  be  made  up  running  55  miles  per  hour,  or  25 
miles  an  hour  faster,  taking  300  -^  25,  or  12  hours.  Hence,  the 
distance  from  5  to  C  is  12  x  55,  or  660  miles,  and  (50  — 12) 
times  30,  gives  1140  miles  from  A  to  B, 

—  From  "  The  School  Visitor." 

203.  Volume  of  sphere  =  2  times  volume  of  double  cone. 
Surface  of  sphere  =  V2  times  surface  of  double  cone. 

204.  20  rods. 

205.  For  bodies  above  the  earth's  surface,  the  weight  varies 
inversely  as  the  square  of  the  distances  from  the  center.  Hence, 
to  weigh  yL  as  much  as  at  the  surface,  the  body  must  be 
Vl6  =  4  times  as  far  from  the  center,  or  16,000  miles,  and  the 
required  height  above  the  surface  is  16,000  -  4000  =  12,000 
miles. 

206.  1  mile.  207.    7.2  inches.  208.    34. 

209.  The  difference  between  the  bank  and  the  true  discount 
is  always  the  interest  on  the  true  discount.  Hence  S9  is  12% 
of  the  true  discount,  which  is  $  75.  The  bank  discount  is  $  9 
more,  or  S84,  which  is  12  %  of  the  face  of  the  note,  and  then 
$84  divided  by  .12  gives  $  700,  the  face. 

210.  By  the  prismoidal  formula,  the  volume  F  is  |-  of 
(upper  base  +  lower  base  +  4  times  middle  section)  x  length. 
Therefore  F=  i  (4  x  4  -f  2  x  3  +  4  X  3  x  3J)  x  120  -- 144  =  8f 
feet,     Ans. 

211.  Place  the  box  on  its  end  and  put  in  11  rows  of  5  and  4 
balls,  alternately,  making  a  total  of  50  balls  in  the  first  layer. 


ANSWERS   AND  SOLUTIONS  181 

Place  the  second  layer  in  the  hollows  of  the  first,  and  it  has  6 
rows  of  4  each  and  5  rows  of  5  each,  making  49  balls  in  the 
second  layer.  In  this  manner  12  layers  may  be  placed,  making 
a  total  of  (50  +  49)  x  6  =  594  balls. 

—  From  "  The  Ohio  Teacher." 

212.  If  the  field  were  48  feet  wide,  it  would  take  one  post 
less  at  each  end  and  two  less  at  each  side,  or  6  less ;  but  to 
make  66  less,  the  field  must  be  11  x  48  =  528  feet,  or  32  rods 
wide,  and  64  rods  long ;  area,  12.8  acres. 

213.  17.584,  specific  gravity. 

214.  7^  feet,  the  distance  the  ball  bounds.  30  feet  equals 
the  whole  distance  the  ball  moves. 

215.  Let  r  =  rate  per  month,  12  r  =  rate  per  annum,  p  =  sum 
borrowed,  n  =  number  of  payments,  q  =  cash  payment.  Then, 
from  algebra,  we  get 

^  =  0^^>  9  =  9J,  p=$500,  n  =  72. 

•••  to  -Pr){^  -  ry  =  q,  and  (19  -  IQOO  r)(l  +  r)"  =  19. 
.-.  r  =  .00911,  and  12  r  =  .10932  =  10.932  %. 

216.  1178.1  square  feet.  218.   6.864+  inches. 

217.  5JJJ  ounces.  219.   72  and  96. 

220.  Since  the  numbers  have  a  common  factor  plus  the  same 
remainder,  if  the  numbers  are  subtracted  from  one  another,  the 
results  will  contain  the  common  factor  without  the  remainder, 
thus : 

364  414  539 

364 414 

50  125 

The  largest  number  that  will  divide  all  of  these  numbers 
is  25,  Ans. 

221.  I.  Let  S  =  selling  price  and  (7=  cost. 


182  MATHEMATICAL   WRINKLES 

S  —  C 

Then,  S  —  C=  gain  and  —  =  rate  of  gain. 

O 

Also,  S  —  -f^-^  C  =  supposed  gain,  and 

J§ 9  2     (J        1_0_0_  g Q 

— gJ^J, —  =  ^^    „ =  supposed  rate  of  gain. 

.-.  S- 0=1^0,  or  15%,  ^ns. 
11.  A  jSJiort  Solution.     8  :  10  =  92  :  115. 
115  - 100  =  15  %  gain. 

222.  Let  |-  =  distance  the  hour  hand  moves  past  3  o'clock. 
^  =  distance  the  minute  hand  moves  in  the  same  time. 
Then  -2^  + 1  =  -^i  =  distance  they  both  move. 

But  the  distance  they  both  move  =  45  minutes. 
.-.  -^  =  45  minutes. 

1  =  ^  of  45  minutes  =  l^f  minutes. 
-2/-  =  24  X  l|f  minutes  =  41/^  minutes. 
.-.  It  is  41  j7^  minutes  past  3  o'clock. 

223.  The  weight  of  the  first  ball  is  3|  times  an  equal  bulk  of 
water,  and  that  of  the  second  is  2^^  times  the  equal  bulk  of 
water;  hence,  3|- times  the  volume  of  the  first  equals  2^  times 
the  volume  of  the  second  ball.  But  the  volumes  vary  as  the 
cubes  of  the  diameters ;  hence,  the  required  diameter  is, 

d  =  -^"(3f  -i-  210)  =,  1^  feet,  Ans. 

224.  The  amount  of  $500  for  2  years  at  6%  is  $560; 
$  2500  —  $  560,  or  $  1940,  is  the  amount  of  the  note,  the  pres- 
ent worth  of  which,  for  24  -  8,  or  16  months,  is  $  1796.30. 

225.  The  present  worth  of  $201  for  30  days  at  6%  is  $200; 
the  present  worth  of  $224.40  for  4  months  at  6%  is  $220. 
Hence,  the  present  rate  of  gain  is  (220  -  200)  --  $200  =  10%, 
Ans, 


ANSWERS  AND  SOLUTIONS  183 

226.  If  the  65  minutes  be  counted  on  the  face  of  the  same 
clock,  then  the  problem  would  be  impossible,  for  the  hands 
must  coincide  every  65^-^  minutes  as  shown  by  its  face,  and  it 
matters  not  if  it  runs  fast  or  slow ;  but  if  it  is  measured  by 
true  time  it  gains  j\  of  a  minute  in  (yo  minutes,  or  yY^  ^^  ^ 
minute  per  hour,  Ans. 

227.  The  loss  of  weight  of  an  immersed  body  equals  the 
weight  of  the  fluid  displaced.  Hence  970  —  892  =  78  ounces, 
weight  of  water  displaced,  and  970  —  910  =  60  ounces  weight  of 
alcohol  displaced.  But  as  water  is  taken  as  the  standard  of 
comparison,  the  specific  gravity  of  alcohol  is  60  -r-  78  =  ^§  = 
.769+,  Ans. 

228.  The  rate  of  the  ship  is  J*  per  hour,  while  that  of  the 
sun  is  15°.  When  they  both  move  west,  the  sun  gains  14|° ; 
but  when  the  ship  moves  east  the  sun  gains  15J°.  Therefore 
since  the  sun  must  make  a  gain  of  360°  in  each  case,  the  time 
from  noon  to  noon  is  360°  -5-  14|  =  24^^  hours,  west  and  360°  -5- 
15  i°  =  23^  hours  east. 

229.  ^  of  165  acres  =  65  acres,  the  amount  of  land  each  man 
should  furnish. 

100  acres  —  55  acres  =  45  acres,  the  number  of  acres  A  fur- 
nishes C. 

65  acres  —  55  acres  =  10  acres,  the  number  of  acres  B  fur- 
nishes C. 

Hence,  ^  ot  $  110  =  $90,  the  amount  A  should  receive,  and 
^  of  $  110  =  $  20,  the  amount  B  should  receive. 

230.  Let  r  be  the  internal  radius  of  the  cup ;  and  the  volume 
of  a  quart  of  wine,'  57f  inches.  Then  240  ttt^  -t-  (3  X  57J)  = 
value  of  wine  in  cents. 

Also  40  ttt*  =  value  of  cup  in  cents. 

.  A(\^       2407rr» 
••^^'^  =  3-^^- 

.-.  r  =  28  J  in.     .-.  40  irr*  =  $  1047.74,  Ans. 


184 


MATHEMATICAL   WRINKLES 


231.  11  times. 

232.  Eleven  integral  solutions,  as  follows : 


Average  price  =  1 


SI 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

91 

82 

73 

64 

55 

46 

37 

28 

19 

10 

1 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

233.  The  volume  is  found  by  the  prism oidal  formula. 

1 Z  (2  X  2  4- 1  X  1 2  +  4  X  f  X  V )  - 144  =  ^  Z  feet,  or  if  I  be 
the  length  in  feet,  the  board  measure  is  ||  of  the  length  in 
feet. 

234.  l|i  board  feet. 

,  235.    Since  .0^  ==  .05,  .^  must  be  .5. 
236.   96  acres. 


7|-  acres. 


237.  40  rods ;  30  rods  ; 

238.  A  liter  of  ice  weighs  918  grams  and  a  liter  of  sea  water 
weighs  1030  grams.  Then  918  divided  by  1.03  equals  891.262 
cubic  centimeters  displaced  by  one  liter  of  ice,  and  1000  — 
891.262  is  108.738  cubic  centimeters  above  water.  Now  108.738 
divided  by  1000  gives  .108738  of  the  whole  above  water,  and 
700  divided  by  .108738  equals  6437.5  cubic  yards,  the  volume 
of  the  iceberg. 


Side-wall^^-^'^ 

1 
Ei^d 

s^ 

^-^ 

1 

End 

Floor 

239.  The  distance  SFis  the  hypotenuse  of  a  right  triangle  = 
V(15)2  +  (39)2  ^  41.78+  ft. 

240.  $  563.23  due  A. 

241.  (a)  The  least  time  required  is  59f|  seconds  past  12. 
(6)  The  least  time  required  is  30y\^2T  seconds  past  12. 
(c)  The  least  time  required  is  1^%V  minutes  past  12. 


ANSWERS  AND   SOLUTIONS  185 

242.  S  2500.00. 

243.  S225  =  first  payment;  $675  =  second  payment. 

244.  First,  $8400;  second,  S7800;  third,  $7280. 

245.  8  yards  of  first  kind;  16  yards  of  second  kind. 

246.  21^  minutes  past  4  o'clock. 

247.  A,  261^  days;  B,  120  days. 

248.  10.  251.   2  inches. 

249.  7ifeet;  8|  f eet.  252.   2  cones. 

250.  13,066.4  miles. 

ALGEBRAIC  PROBLEMS 

1.  Let  X  =  your  age. 

y  =  difference  between  our  ages. 
Then  a;  -h  y  =  my  age. 

and  (x-^y)-\-(x-\-2y)  =  100. 

Solving,  X  =  33^  and  x-}-y  =  44 J. 

2.  A,  72  hours ;  B,  90  hours. 

3.  Let  the  time  be  x  minutes  past  10  o'clock.  We  assume 
that  at  the  beginning  of  every  minute  the  second  hand  points 
at  12  on  the  dial.  The  distance  of  the  second  hand  from  the 
minute  hand  at  the  required  time  is  60  a;  —  a;  =  59  a; ;  and  that  of 
the  second  hand  from  the  hour  hand  is 

60  -  60  a;  -  (10  -^  tV  x)  =  50  -  60  X  -  -jJj  a;. 

.-.  59  x  =  51  -  60  X  4-  iV  ^• 
Solving,  x  =  y^:j^  minutes  =  25| Jff  seconds. 

—  From  "  American  Mathematical  Monthly." 


186  MATHEMATICAL   WRINKLES 

4.  Let     s  =  distance    between   cars    going    in    the    same 

direction. 
Let  i  =  interval  of   time  between  cars  going  in  the 

same  direction. 
Let  X  =  rate  of  car. 

Let  y  =  rate  of  man. 

Then,  x  —  y  =  rate  of  approach  when  both  travel  in  the  same 

direction. 
x-\-y  =  rate  of  approach  when  they  travel  in  opposite 

directions. 
By  conditions  of  problem, 

12(x-y)=^s  =  4.{x  +  y). 
.'.x  =  2y. 

Also,  ^  =  ^  =  ii^±l)=6. 

X  X 

Therefore  the  interval  between  cars  is  6  minutes,  and  my 
rate  is  half  the  speed  of  the  cars. 

—  F.rom  "  School  Science  and  Mathematics." 

5.  Let       X  =  the  number  of  eggs  for  a  shilling. 

Then  -  =  the  cost  of  one  Qgg  in  shillings. 

X 

12 

and  —  =  the  cost  of  one  dozen  in  shillings. 

X 

12 

But  if  X  —  2  —  the  number  of  eggs  for  a  shilling,  then  - 

X  —  2 

would  be  the  cost  of  one  dozen  in  shillings. 

12        12      1 

.*. =  -—  (1  penny  being  y^  of  a  shilling). 

X  —  2      X      12 

Solving,  a;  =  18  or  — 16.     Then  if  18  eggs  cost  a  shilling,  1 

dozen  will  cost  if  of  a  shilling,  or  8  pence,  Ans. 

6.  Let  X  =  amount  per  yard  received  by  one. 
Then               ic  -j- 1  =  amount  per  yard  received  by  the  other. 


ANSWERS   AND   SOLUTIONS  187 

Solving,  «=  1.7808. 

One  builds  56.15  yards  at  $1.7808;  other  builds  43.85  yards 
at  $2.2808,  Ans. 

7.  1760  yards,  or  1  mile. 

8.  Let  X  =  number  of  acres. 

160  X  =  number  of  dollars  for  which  the  land  sold. 
Then  since  1^  inches  =  diameter  of  a  dollar, 

1^(160  x)  =  240  X  =  perimeter  of  square  in  inches. 

-y  or  60  a:  =  length  of  one  side  of  the  square  in  inches. 

4 

-— ^,  or  5  a;  =  length  of  one  side  of  the  square  in  feet. 
(5  xy  =  the  number  of  square  feet  in  the  square. 


=  number  of  acres. 

25  X* 

•  •  43560     "• 

Solving, 

x  =  1742.4,  ^715. 

9.   2652.5+  feet 

;  2627.4-^  feet. 

10.   Let 

X 

=  one  side  of  the  square  in  feet. 

Then 

43560 

=  the  number  of  acres. 

16aj  = 

=  the  number  of  feet  of  boards  in 

the  fence 

16  X 
11 

=  the  number  of  boards  in  the  fence. 

a? 

16x 

**' 

43560 

11 

Solving, 

X: 

=  63,360. 

Then 

X^ 

43560 

=  92,160  acres,  or  144  sections. 

11.   10^ 

hours. 

188  MATHEMATICAL   WRINKLES 

12.  Let  X  =  rate  of  faster  train  per  hour  in  miles. 

y  =  rate  of  slower  train  per  hour  in  miles. 
In  5  seconds  both  trains  travel  600  feet. 
.-.  in  one  hour  they  travel  81^  miles,  or 

x  +  y  =  Sl^\.  (1) 

In  seconds  the  fast  train  gains  600  feet. 

.'.  in  one  hour  the  fast  train  gains  27 ^^  miles,  or 

x-y=2T^.  (2) 

Solving,  X  =  54y6_,  and  y  =  27 ^^y,  Ayis. 

13.  Let  X  =  number  of  minutes  until  6  o'clock. 
Then  6  hours  —  x=  time  past  noon. 

3  hours  -f-  4  a;  =  time  past  noon  50  minutes  ago. 
.♦.  360  -  .T  =  180  +  4  a;  +  50. 


Solving, 

X  =  26,   Ans, 

14.   Let 

X  =  cost  of  the  gun  in  dollars. 

-— -  =  per  cent  of  loss. 
lUU 

Then 

(^y.=ioss. 

x'        ^ 
100 

Solving, 

a;  =  90,  or  10. 

••• 

$  90,  or  $  10  =  the  cost  of  the  gun. 

15.    Let 

X  =  number  of  eggs  he  brought. 

Then 

a;  +  1  =  1  of  them. 

and 

f  (a;  +  1)  =  number  of  eggs  in  the  nest. 

Also 

a;  —  2  =  i  of  them. 

and 

2  (a;  —  2)  =  number  of  eggs  in  the  nest. 

...2(x- 2)  =  1(0^  +  1). 

Solving, 

aj  =  ll. 

Then 

2  (a; -2)  =  18,   Ans. 

16.   Let 

X  =  cost  of  lot  in  dollars. 

ANSWERS   AND  SOLUTIONS  189 


X 

100 

=  per  cent  of 

gain. 

) 

100 

=  gain. 

^  +  100  = 

=  144. 

Then 


Solving,  X  =  80,  or  -  180.  $  80  =  Arts. 

17.   6  inches.         18.    VS.  19.    |  ±  jVS   and  J  ±  hV5. 

20.  21  minutes  49^  seconds  past  4  o'clock. 

21.  Let    X  =  number  of  men  in  a  side  of  the  first  square. 
Then        xr  =  number  of  men  in  the  first  square, 

and     ic^  +  39  =  number  of  men. 

Also  X  -h  1  =  number  of  men  in  a  side  of  the  second  square 

Then     (x  -f  1)^  =  number  of  men  in  the  second  square, 
and  (a;  + 1)*  —  60  =  number  of  men. 

...  (x  4- 1)'- 50  =  ar^-f  39. 

Solving,  x=  4A. 

Then     a:*  4- 39  =  1975,   Ans. 

22.  12  cents.  23.   4  feet. 

24.  iV5  and  i(VE  ±  5), 

25.  16.  26.   6  feet. 

27.    Let  X  =  cost  of  first  horse. 

80  —  a;  =  cost  of  second  horse. 
Then  80  —  a;  =  gain  on  first  horse. 

and      80  —  (80  —  x)  =  gain  on  second  horse. 
80-a; 


x 

X 


=  rate  of  gain  on  first  horse. 


80 -a; 
80 -X     1 


=  rate  of  gain  on  second  horse. 


80-a;  X  5 

Solving  a;  =  41.905, 

and  80 -x  =  38.005. 


190  MATHEMATICAL   WRINKLES 

28.  The  lot  is  100  feet  x  100  feet  =  10,000  square  feet.  The 
house  and  the  driveway  each  covers  5000  square  feet. 

Let  X  =  the  width  of  the  driveway.  On  each  side  of  the 
lot  it  extends  from  front  to  rear  100  feet ;  total,  200  x  square 
feet.  The  house  is  100  feet  —  2  x  feet ;  the  driveway  behind 
the  house  is  100  feet  —  2x  feet  long  by  x  feet  wide ;  the  total 
number  of  square  feet  is  100  x  —  2  x^.  The  total  number  of 
square  feet  in  the  driveway  (at  sides  of  lot  and  rear  of  house) 
is  200 x  +  lOOx-2  x\ 

.'.  -  2  a;2  4- 300  a?  =  5000. 

Solving,  x  =  19.1. 

29.  672. 

30.  7.416  inches  from  either  end. 

31.  Their  monthly  wages  may  be  any  number  of  dollars. 
If  they  receive  more  than  $  50  a  month  they  will  each  lay  up 
the  same  sum.  If  they  receive  less  than  $50,  they  will  be- 
come equally  indebted. 

32.  1 

33.  2(l-}-x')=(l-j-xy. 

2-\-2x^  =  x^-j-4:x'  +  6x'-{-4x  +  l. 
Transposing, 

aj4_4^_6^_  4.x -{-1  =  0. 
Adding  12  x^  to  both  sides, 

x*-4.a^  +  6x^-4.x  +  l  =  12x^ 
But  x^-4:a^-\-6x'-4.x-\-l  =  (x-iy. 

Then  (x-iy-12x'  =  0, 

or  (x^  -2x^iy-12x'=0. 

Factoring, 

(aj2_2a;  +  l-f  2cc-\/3)(ic2-2a^  +  l-2a;V3)  =  0. 
...  a^_  2a;  +  l  +  2a;\/3  =  0, 
and  x'-2x-^l-  2  a;V3  =_  0. 

Solving,  »  =  1  -  V3  ±  V3  -  2  V3, 

or  1  +  V3  ±  V3  +  2  V3. 


ANSWERS  AND   SOLUTIONS  191 

34.  X*  -^  4:  m^x  —  m*  =  0. 
Factoring, 

(a^  4-  mxy/2  -  mV2  +  m*)(aj*  -  mxy/2  +  m'y/2  +  m*)  =  0. 
.-.  x^  ^_  wa;V2  -  w2V2  +  m*  =  0. 

o  I   •  w    .  mV2V2-l 

Solving,  x  = =:  ± — 

V2  V2 

35.  a^  +  3/=ll.     (1) 
y2-fx=    7.     (2) 

(3)  y  -  2  =  9  -  ar^,  from  (1). 

(4)  /  _  4  =  3  -  a:,  from  (2). 

(5)  2/  -  2  =  -^ ^,  from  (4)  by  dividing  by  y  +  2. 

'  2/  +^    y -^ ^ 

3  a; 


•.  9  -  ar^  = 


2/  +  2     y  +  2 

a^ ^,  by  transposing. 


2/  +  2  2/  +  2 

Then 

^"7+2"^V27T4)  ='^"^rf2"^(,27T4j' 

completing  the  square. 

3 =  x ,  extracting  the  square  root. 

22/ +  4  2^  +  4' 

Then  canceling,  a;  =  3, 

and  substituting,  y  =  2. 

Note.— From  Horner's  method  we  find  a;  =- 2.803,  3.681,-3.778. 
Hence  y  =  3.131,  -  1.849,  -  3.283. 

36.  x  =  2;  2/=l.  39.   a:  =  4;  2/ =9. 

37.  a;  =  4;  2/ =  6.  40.   a;  =  4;  y  =  9. 

38.  a;  =  2 ;  2/  =  3. 

41.  x  =  ±2,  ±-^;2/  =  ±5,  ±-^;2  =  ±3,  ±-l3. 

V7  Vi  V7 

42.  52/(a^  +  l)-3ar3(2/»  +  l)  =  0.     (1) 

15/(ar'  +  l)-x(2/«  +  l)=0.     (2) 


192  MATHEMATICAL  WRINKLES 


(3)    lB(^^yl±l,iror.(2). 

(5)  5(^-^^\  =  s(y-^  ^\  from  (4). 

(6)  15('aj  +  ^')  =  2/'  +  -3,from(3). 


a;y      SV         2/^ 


(8)       3(x  +  ±]  =  ^(/-fA),  from(6). 


^  (6). 


adding  (5)  and  (6) 
Then 


\/^{x  -\-  -)  =  2/  +  -,  extracting  the  cube  root. 

\      ^J  y 


But  since  y-{--  =  7?-\-  — 

y  ^ 

Dividing  by  a;  +  -  ?  we  have 
a; 

a;"  +  2  +  ^  =  •^+  3,  by  adding  +  3. 


ANSWERS  AND   SOLUTIONS  193 

*  +  -  T=  V  \/5  +  3,  by  extracting  the  square  root. 
Also  ^  -  -  =  ^y/'5  -  1,  by  subtracting  1. 


•••  ^  =  i[V^5  +  3  4-  Vs/5-1]. 

But        y  +  l  =  (^^-^l)</5  =  <^^ww+s. 
y-  -  y(v'r) .  V</5  -f  3)  4-1  =  0. 

y  =  iC^^  •  Vs^/'s  +  3  ±  V</25(\/5  +  3)-4]. 
.-.  2/  =  J [^5  .  V^5  4.3  ±  Vl  +  3</25]. 

43.  Let  a;  =  number  of  feet  in  one  side  of  the  field. 
Then,  x^-{x-  mf  =  (x  -  66)1 

Solving  X  =  225.3356+  feet. 

.-.  he  had  50776+  square  feet,     Ans, 

44.  2.93+  gallons. 

45.  Let  X  be  the  least  integral  number  that  will  satisfy  the 
conditions ;  then  we  shall  have 

a;  =  39y4-25  =  252  +  19  =  19ri  +  ll; 
whence,         z  =  y4--^y-\-  ^%. 

For  integral  values  y  must  be  17,  42,  67,  92,  117,  142,  167,  etc. 
Hence  the  value  of  y  that  will  satisfy  the  last  value  of  x  in 
third  line,  is  167 ;  then 

a;  =  39  y  -h  25  =  25  2  +  19  =  19  n  -h  11  =  5369. 

—  From  "  American  Mathematical  Monthly." 

46.  Dr.  A  saves  f;  Dr.  B  ^;  and  Dr.  C  |^.     Hence,  the 
chance  for  one  who  is  dosed  by  all  three  is 

47.  Let  a;  =  Richard's  age  and  y  Robin's  age. 
Then  2x-y4-a;  =  99, 

and  2(2y-'X)=2x-y. 

,\  x  =  45  and  y  =  36. 


194  MATHEMATICAL   WRINKLES 

48.  If  the  assertion  is  true,  A  and  B  tell  the  truth  and  C  is 
mistaken.     The  chance  of  this  is 

3"  X   Y  X  -g-  —  y  0^3". 

If  the  assertion  is  not  true,  A  and  B  are  mistaken  and  C 
tells  the  truth.     The  chance  of  this  is 

1    V    1    V    4   —      4 
-3    A   y   A    5-  —  y  0-3-. 

Now,  the  assertion  is  true  or  it  is  not  true,  and  12  chances 
are  in  favor  of  its  being  true  to  4  in  favor  of  its  being  not 
true.     Hence,  the  probability  of  its  truth  is  if  or  |,     Ans. 

—  From  "  The  School  Visitor." 

49.  Let  X  =  weight  of  the  plank,  acting  through  its  mid- 
point with  lever  arm  7,  while  the  weight  of  196  has  the  lever 
arm  1 ;  the  equation  of  moments  is : 

7x  =  196.     .-.  X  =  28. 

50.  Let  X  =  number  at  5  cents,  y  =  number  at  1  cent,  z  = 
number  at  |  cent.     Then 

X  -{-  y  -{-  z  =  100  =  5x-\-y-\-lz. 
Eliminating  z  we  get  the  indeterminate  equation,  9x-\-y=100. 

.'.  y  =  100-9x. 

This  equation  gives  us  eleven  integral  solutions,  as  follows : 

aj  =    1,    2,    3,    4,    5,    6,    7,    8,    9,  10,  11. 

y  =  91,  82,  73,  64,  55,  46,  37,  28,  19,  10,    1. 

z=    S,  16,  24,  32,  40,  48,  56,  64,  72,  80,  88. 

51.  9f.  54.    a +  5. 

52.  6.  55.    l  +  V2-f  V3. 

53.  17  years,  2  months,  2  days.         56.   1  +  V|  —  V|. 
57.   x~i  +  x~^  =  6. 

Solving  for  x'^,  x-^  =  -^±  V6Ti  =  2,  or  -  3. 
.-.  a;  1  =  16,  or  8L 

.-.    a?  =  yg-,    or   -gy. 


ANSWERS  AND   SOLUTIONS  195 

58.  64;  (-33)1  62.   x  =  Q,  -^, 

59.  a,  ~  6.  '  :y  =  12,  -  9. 

60.  «  =  !,}.  ^  

y  =  f,|.  63.    a:=Vi(V2-l), 

2=  ±2.  1 


61.   x  =  ^(l±V3),  ^2(V2-1) 

id 


|(i±-^),  64.  x  =  fVio±yvg, 

j,  =  |(lW3),  y  =  |V2±|V5. 


66.  x  =  |[V-J/3  +  3+V-J/^_l], 


66.  600  yards. 

67.  5(„_l)(2n-l)  yards. 
o 

68.  6  minutes. 

69.  Silenus  in  3  hours ;  Dionysius  in  6  hours. 

70.  Let  X  =  time  required  to  overtake  B.     He  travels  20-|-2a; 

20  -^  2  3* 
miles.     Hence  — — —  =  his  rate.     Let  y  —  time  to  go  from 


B  to  A.     He  travels  2  V(100  +/)  miles.     '^^^^^^'A  =  his 

rate.     After  reaching  B  a  second  time  he  has  left  10  —  a:  —  2  2/ 
hours  to  go  2  a;  -}-  4  y  miles. 

. .      ix-\-\y    __  j^.g  j.^^      Tg^^  j^^g  ^.^^g  ^g  uniform.     Hence 
10-a;-22/ 
we  get 

20  +  2x^2V(100+^^„^g^^gy,^^^  (1) 


196  MATHEMATICAL  WRINKLES 

20  4-2a;        2  a;  4-4?/  9.0  ^?/^      ^r,        /^s 

-^  =  103^'°'^  ^  +  20^  =  50-102'     (2) 

li  x  =  vy  in  (1),  we  get  5v^ —  v  =  5  ot  v  =  1.10499. 
.*.  X  =  1.10499  y.     Substituting  in  (2)  gives 

14.41996  /  +  10  2/  =  50,  ovy  =  1.54737. 
.-.  X  =  1.70983.     Rate  =  ^  -f  2  =  13.69707  miles  an  hour. 

X 

From  "  The  American  Mathematical  Monthly." 

First  Solution  : 

71.   Let  X  =  the  part  of  a  man's  work  the  boy  does. 

k  =  the  number  of  bushels  of  apples  the  man  shakes  off  in 
a  day. 

kx  =  the  number  of  bushels  of  apples  the  boy  shakes  off  in 
a  day. 

-^  =  the  number  of  bushels  of  apples  each  man  picks  up  in 
o 

a  day. 

ka^ 

— -  =  the  number  of  bushels  of  apples  the  boy  picks  up  in  a 
o 

day. 

.  2kx      kx^  ^4:k 

**    3  3         5  * 

.'.  x  =  0.843909,  about  ||  or  i^f. 
Boy's  share  of  pay  =  $10,976;  each  man's  share  =  $13.01. 

Second  Solution  : 

Suppose  a  man  does  x  times  as  much  work  as  a  boy. 

By  the  first  condition, 

Shaking  the  apples :  picking  them  up  =  1 :  3  a;. 

By  the  second  condition, 

4  X 
Shaking  the  apples  :  picking  them  up  =  -—  :  2  o^  +  1. 

o 

.-.  1 :  3  a;  =  —  :  2  a;  +  1,  or  12  a;2  -  10  oj  -  5  =  0. 
5 


ANSWERS  AND   SOLUTIONS  197 

.-.  X  =  1.185. 

.'.  The  money  is  divided  into  parts  proportional  to  1,  1.185, 
1.185,  and  1.185. 

.-.  The  boy  receives  $10.98  and  each  man  receives  $13.01. 
—  From  "  School  Science  and  Mathematics.** 


GEOMETRICAL   PROBLEMS 

1.  Construct  the  AABCy  whose  base  ^B=  sum  of  parallel 
sides,  Z  C=  angle  between  diagonals  and  where  AC-\-CB  = 
sum  of  diagonals. 

Take  the  point  E  on  AB^  such  that  CE=CB;  from  E  draw 
EF  parallel  and  equal  to  AC  meeting  CB  in  0;  join  B  and  F. 
CFBE  is  the  required  trapezoid. 

Proof:  AE=CFy  hence  EB -\- CF  —  ^ly^n  sum  of  parallel 
sides.  Since  EF  =  AC,  EF-^  CB=  given  sum  of  parallel 
sides.    And,  finally,  Z  EOB  =  Z  ACB. 

2. 

Let  r  =  the  radius  of  the  three  equal 
circles. 

Then  2  r  =  the  diameter. 

(1)  The  area  of  a  semicircle  whose 

radius  is  r  =  -— • 

(2)  The  area  of  the  A  is  ^  +200. 

(3)  But  the  area  of  the  equilateral  A  is  r*  V3. 
.-.  i?\/3  =  ^+200. 

Solving,  r  =  498.06  feet.     Then  2  r  =  996.12  feet 

3.   6J  feet.  4.   60  feet. 

5.  Let  R  =  the  radius  of  the  sphere,  and  2  h  the  altitude  of 
the  cylinder.     Then  R  —  h  =  the  altitude  of  the  segment  of  the 


198  MATHEMATICAL   WRINKLES 

sphere,  and  ^(R^  —  h^)  is  the  radius  of  the  base  of  the  seg- 
ment and  the  radius  of  the  cylinder. 
The  volume  of  the  2  segments 

=  2  [i  7r(JR  -  hf  +  i  ,r(i2  -  Ji)  {R'  -  7i2)], 

and  the  volume  of  cylinder  =  2  irh{R^  —  h^). 

.'.  ^7r(R^  —  h^)  =  the  volume  of  the.  segments,  and  the  cylin- 
der =  -|(|  ttR^),  by  the  conditions  of  the  problem. 

.■.3¥=  RK  .-.  1 7r{R^  -  ^')  =  f  7r7i^  =  600,  by  the  conditions 
of  the  problem. 

3 


.-.  2/i  =  2V(225/7r). 

6.  1012+ square  feet.  11.  An  isoceles  right  triangle. 

7.  99.379  feet;  11.119  feet.  12.  6.18  rods. 

8.  108.046  feet.  13.  3904. 

9.  15.708  feet.  14.  3  feet. 
10.   13.4316  feet. 

15.  Upon  the  given  base  AB  construct  a  circle  whose  segment 
ACB  shall  contain  the  given  vertical  angle.  Through  E,  the 
mid-point  of  AB,  draw  EF  perpendicular  to  AB,  meeting  the 
circumference  at  F.  Join  FB,  and  perpendicular  to  FB  draw 
BG  equal  to  ^  the  given  bisector  of  the  vertical  angle.  With 
Q  as  center  and  BG  as  radius  describe  the  circle  BHL,  and 
draw  FGL.  With  F  as  center,  FL  as  radius,  describe  a  circle 
cutting  the  given  circle  in  C  Join  FC,  cutting  AB  in  D. 
Then  ABC  is  the  triangle  required. 

In  the  triangles  FCB  and  FBD,  ZFCB  =  ZFBA,  since  arc 
^F=arc  FB  ;  also  Z  CFB  is  common,  hence  the  triangles  are 
similar,  and  FC:FB  =  FB:FD;  but  FL(=FC)  :  FB  =  FB  : 
FH.     Therefore  FH=  FB  and  HL  =  CD. 

Hence  in  the  triangle  ABC,  AB  is  the  given  base,  Z  ACB 
the  given  vertical  angle,  and  CD  the  given  bisector,  and  the  tri- 
angle is  satisfied  in  every  condition. 

—  From  "  American  Mathematical  Monthly." 


ANSWERS   AND   SOLUTIONS  199 

j 

16.  f  18.    ^. 

17.  Height  =  radius.  19.    12.91  miles. 

20.  Let  x  =  a.  side  of  the  equilateral  triangle.  Also  a,  6, 
and  c  =  the  given  distances  from  the  point  to  the  sides. 

x^ 

(1)  The  area  of  the  equilateral  A  =  —  V3. 

(2)  The  area  of  the  equilateral  A  =  ^  (a  -f-  &  -f  c). 
.•.^V3  =  ?(a  +  6  +  c).  Solving,  «  =  «±^. 

21.  10341.1  cubic  inches.  22.    16.9704. 

23.  Three  inches  solid  is  greater,  for 
Three  solid  inches  =  3  cubic  inches. 
Three  inches  solid  =  27  cubic  inches. 

24.  Their  homes  will  be  the  vertices  of  an  equilateral  tri- 
angle, and  hence  the  well  must  be  dug  where  the  bisectors  of 
the  angles  meet. 

26.  600  square  feet.  30.   10  feet. 

27.  8.0558.  33.   39.79  cubic  inches. 
29.    7  feet. 

34.  Diameter  of  the  fixed  circle. 

36.  4  feet.  44.  769.421  square  feet. 

36.  10  feet.  45.  936.564  square  feet. 

37.  4330.13  square  inches.  46.  1119.615  square  feet. 

38.  10,000  square  inches.  47.  2^  feet. 

39.  17204.77  square  inches.  48.  16|  inches. 

40.  259.81  square  feet.  49.  1.755  inches. 

41.  363.39  square  feet.  50.  10.198  feet. 

42.  482.84  square  feet.  61.  10.863  inches. 

43.  618.182  square  feet.  62.  10.142  ft. 


200 


MATHEMATICAL  WRINKLES 


53.  331  feet. 

54.  71.344  feet  from  smaller;  70,071  feet  from  larger. 


55.  15.38756  feet. 

56.  2  feet. 

57.  1.84  cubic  feet. 

58.  126^^  square  inches. 

59.  122.84  square  inches. 

60.  11.66  square  inches. 

61.  251.328  square  feet. 

62.  6.8068  cubic  feet. 


63.  4.192  feet  from  large  end. 

65.  154.9856. 

67.  2106  square  yards. 

68.  80=base;  60  =  altitude. 

69.  101  feet. 

70.  29.41  rods. 
73.  78.572  feet. 


74.  Let  a  =  BC  and  AC,  the  ladders  ; 
J57)=10;  and  JBi^=7;  then  ^Z)=a-10. 
Let  X  =  BE ;  then  AE  =  14  —  a; ;  and 
we  have  V(ED''  +  EA')  =  a  -  10.  But 
ED'  =  10^ -a;^  and  EA"  =  (14  -  xf. 
Hence  V[102  -x^-^  (14  -  xf]  =  a  -  10. 

Also,  7  :  ic  =  a :  10,  whence  ic  =  70  ^  a. 
Substituting  and  reducing, 

a3_  20  a^- 196  a +  1960  =  0; 

.-.  a  =  24.72189  feet,  Arts. 

75.  78  feet.  80.    13.65  rods. 

81.  Let  PA'  and  B'C  intersect  at  K.  Draw  through  K  a 
line  parallel  to  BC  cutting  AB  at  x  and  AC  at  y.  A  PB'C  is 
isosceles  ;  therefore  Z.  B'  =  A  C.  The  points  P,  K,  y,  B'  are 
concyclic ;  hence  Z  y  =  Z  B'.  The  points  P,  x,  C,  K  are  con- 
cyclic  ;  hence  Zx  =  Z  C  .'.  Zx  =  Zy,  and  A  Pxy  is  isosceles. 
Hence  K  is  the  mid-point  of  xy,  since  PK  is  perpendicular  to 
xy.     .:  Kis  on  the  median  through  A. 

—  From  "School  Science  and  Mathematics." 

82.  In  A  ABC  let  m  be  the  bisector  of  the  Z  A,  and  n  the 
bisector  of  Z  B. 


ANSWERS  AND  SOLUTIONS  201 

Then, 


2 


m  = V6  •  c  •  s(«  —  a),  and  n  = Va  •  c  • « (s  —  6). 

See  Schultze  and  Sevenoak's  "Geometry,"  page  J 58. 


Vft  •  c  •  8(s  —  a)  = Va  •  c  •  s(.s  —  6), 


ft+c  a+c 

or  b{8-a)^ai8-b)  .^. 

(6  +  c)*      (a  +  c)2  ^  ^ 

Replace  s  by  .}(a  +  6H-c),  simplify,  and  factor,  equation  (1) 
becomes  . 

(6-a)[c»4-c2(6  4-a)  +  3a6c  +  a6(6  +  a)]  =  0. 

Since  the  second  factor  cannot  be  zero,  b  —  a  =  0,  and  a  =  b. 

106.   5  inches. 


MISCELLANEOUS  PROBLEMS 

1.  10,945. 

2.  221.995  cubic  inches.     Solve  by  using  the  formula, 

V=  r*V3(Trc  —  2  rV2),  where  e  is  the  edge. 

3.  367  trees.  4.  84.823  square  feet;  63.617  cubic  feet. 
6.  21J  feet.  6.  88f  square  feet.  7.  8f  square  feet. 
8.  436.21  cubic  inches.  9.   362.8167  square  feet. 

10.  Let  r  =  the  radius  of  the  ball.  Then  (?•  —  4)  will  be 
the  radius  of  the  hollow  sphere  inclosed  by  the  shell. 

As  the  volumes  of  spheres  are  proportional  to  the  cubes  of 
their  radii,  the  conditions  of  the  problem  require  that 

r»  _(r  -  4)»=  i  r»,  or  f  r»  =(r  -  4)». 

4 

.-.  r  = —  =  55.79+  inches,  Ans. 

1— yi 


^ 


202 


MATHEMATICAL   WRINKLES 


ff""*^ 

\p 

X 

7     \ 

Fv-^ 


D 


B 


11.   15.29  feet.  12.   64  feet.  13.   18.62  feet. 

14.   The  sparrow  flies  66|  feet,  the  eagle  133i  feet. 

15.   Let  AB=2o;  ^(7=25 V2; 
OB  =  ^^2. 

BK=100-25=:75', 
PO  =  -^BP'^-OW  =  -2/ VM. 
^P=y(V34  +  V2); 

^^  =  ^=|(Vi7  +  i). 
AEPF=  AE'  =  ^p(9 + Vrr). 

ABCD  =  25^  =  625, 
BEPFDC  =  ^1^(9  +  Vrr  -  2)  =  3476  square  feet. 
EK==  100  -  ^^  =  ¥(7  -  Vi7). 
Area  of  segments  PFH  and  P£'/f 

=  ^^  + 1(2  P^  X  ^/i")  =  3252  square  feet. 

Area    AHLK=  |  tt  x  100^  =  23,562  square  feet. 
23,562  +  3252  +  3476  =  30,290  square  feet. 

—  From  "  The  School  Visitor.'' 

16.  In  order  to  overturn  the  cube  it  must  be  revolved  on  a 
lower  edge  until  the  center  of  mass  is  vertically  over  that 
edge,  and  this  will  require  the  lifting  of  the  300  pounds 
through  a  distance  a{-\/2  —  1),  a  being  the  edge  of  the  cube 
against  gravity. 

.-.  the  work   done  =  300  a(V2  -  1)=  124.26  a  foot  pounds. 

Hence,   the   size   of    the   cube   cannot    be    left    out    of    the 

calculation. 

—  From  "  The  American  Mathematical  Monthly." 

17.  34.6785  feet.  18.    4.72  rods.  19.    7.92  rods. 
20.   22.72  feet.  21.   76.394  feet. 


ANSWERS  AND   SOLUTIONS  203 

22.  11.9206;  8.0794;  8.0794;  11.9206. 

23.  18.2948  feet.  27.   1.87+  feet. 

24.  38.5704.  28.    889.337+ 

25.  124.905  feet.  29.   24,630.144  acres. 

26.  11.817  inches.  30.  2765.45  square  yards. 

31.  Two  feet  from  the  end  of  the  log. 

32.  249.03  inches.  34.  108  sheep. 

33.  16.125  square  inches. 

35.  The  rabbit  goes  133^  yards ;  the  hound  goes  166J  yards 

36.  128.  43.    11.34  feet  per  second. 

37.  180.  44.   90  pounds. 

38.  19.8 ;  35.7 ;  44.6.  45.   350.163  square  yards. 

39.  16.2484  cubic  inches.  46.   2467.4  cubic  feet. 

40.  60**.  47.   602.349  cubic  inches. 

41.  113.0976  square  feet.  48.  355.88  square  inches. 

42.  7.2  inches.  49.   6830.47  cubic  inches. 
50.  842.044  square  inches ;  404.318  cubic  inches. 

52.  125.6638  square  feet;  31.4159  cubic  feet. 

53.  1,184,352.628  cubic  inches. 

54.  Any  force  greater  than  202J  pounds   will  draw  the 
wheel  over  the  log. 

55.  19.7392  cubic  feet.  56.  39.4784  square  feet. 

57.  4421.58  square  inches ;  17,686.32  cubic  inches. 

58.  1473.86  square  inches ;  3457.92  cubic  inches. 

59.  372.30  cubic  feet.  61.   .596+  feet. 

60.  27.12  cubic  inches.  62.   36  square  feet. 


204  MATHEMATICAL   WRINKLES 

63.  42|  cubic  feet.  67.    628.32  square  inches. 

64.  226.2  cubic  inches.  68.   400  square  inches. 

65.  7.61|  square  feet.  69.   339.29  square  feet. 

66.  189.8  inches.        70.  A,  44.69828+  mi. ;  B,  86.81897+  mi. 

71.    79.119  feet. 

72.  Let  X,  or  AOB  be  the  given  angle. 
With  0  as  a  center  and  any  radius, 

describe  a  circle. 

Draw  the  secant  AMN,  making  MN 
1^  equal  to  the  radius  of  the  circle. 

(This  can  be  done  only  by  using  a  graduated 
ruler.) 

Join  the  points  0  and  M. 
Then  Z  N=  ^  Z  X 

Proof.  Z  MNO  =  Z  MOK 

Z  AMO  =  Z  N-^  Z  M0N=2  xZN. 

Z  AMD  =  Z  MAO. 

ZN  +  ZMAO  =  ZX. 

.'.ZN=\ZX. 

For  other  solutions,  see  Ball's  "  Mathematical  Recreations." 

73.  For  20  pounds  on  10  arm,  weight  =  ?5_^i5  =  22f . 

20  X  9 
For  20  pounds  on  9  arm,  weight  =  — — —  =  18  • 

22-|  4- 18  =  40f  pounds  or  |  pounds  he  loses. 
I  -  40  =  ^i^,  -3-i^  of  100%  =  1%  he  loses. 

.    — From  "  School  Science  and  Mathematics." 

74.  Let  X  =  pressure  on  B's  shoulder. 

The  moments  about  A's  shoulder  are  5  x  54  down,  and  9  x 
up. 


ANSWERS  AND  SOLUTIONS  206 

.'.  9a;=  5  x  54.     .*.  x  =  30,  weight  on  B's  shoulder. 

In  like  manner  we  may  let  x  =  the  pressure  on  A's  shoulder. 

Then  9  x  =  4  x  54.     .-.  a;  =  24,  weight  on  A's  shoulder. 

75.  Since  the  momenta  of  the  bullet  and  gun  are  equal  in 
magnitude,  7  v  =  -^  » 1400,  whence  v  =  6.26  foot  seconds  and 

E  =  ^^ 7^A26^  =  4.3  =  energy  of  recoil. 
64.2         64.2  ^^ 

Also  1F=  Fs.     .'.  4.3  =  -^F,OT  F=  12.9  pounds. 

76.  Let  w  equal  energy  of  ball,  and  v  its  velocity  on  emer- 
gence.    Then  w  =  1000^  x  —  •     Energy   after  passing  through 

n 

plank  is  1 1«=  ^^ — • 

.-.  v*  =  |  X  10002,  or  v  =  ^12.87  feet  per  second. 

77.  16,956.1  square  feet.         80.   68,948.77  feet. 

78.  213,825.15  acres.  81.   337.5  cubic  feet. 

79.  989.96  feet.  82.   22.386  feet. 

83.  41Jfeet. 

84.  40  rd.  =  length ;  30  rd.  =  breadth ;  7^  acres  =  area. 

85.  80  rd.  =  length ;  60  rd.  =  breadth. 

86.  100  feet.  90.   31,416  square  feet. 

87.  50  rods.  91.   6  feet;  8  feet;  10  feet. 

88.  48  inches.  92.   .80449+. 

89.  36.57  acres. 

MATHEMATICAL  RECREATIONS 

1.  Let  X  =  difference  between  Mary  and  Ann's  ages. 
Then        24  —  a;  =  Ann's  age. 

Therefore  12 -ha;  =  24  —  a;, 
.-.a;  =  6. 
.*.  24  —  a;  =  18,  Ann's  age. 

2.  20  pounds. 

3.  Take  the   six   matches   and  form   a  tetrahedron.     This 


206 


MATHExMATICAL   WEINKLES 


tetrahedron  will  have  four  faces,  each  face  being  bounded  by 
three  matches  which  form  an  equilateral  triangle. 

4.  The  same.  * 

5.  To  find  the  digit  crossed  out  subtract  the  remainder 
from  the  next  highest  multiple  of  nine. 

6.  To  find  the  figure  struck  out  subtract  the  remainder 
from  the  next  highest  multiple  of  nine. 

7. 

12  inches 


4  in. 

.s 

CO 

4  in. 

{ 

.s 

CO 

4  in. 

1 

1 

J 

16  inches 


8.   99|. 


9.  In  forming  the  second  figure  from  the  four  parts  of  the 
first  the  lines  forming  squares  do  not  coincide  exactly,  thus 
seemingly  forming  &^. 

10.   The  blacksmith  was  right,  as  he  had  to  cut  and  weld 
only  three  links. 

13. 

14.  25;  15;  20.  18.    oo   % 

15.  40  feet  19.   34|;  31f 

16.  Kides  3  :  walks  1. 


21.   None. 


20.    There  will  be  no  lot,  since  the  given 
dimensions  will  not  make  a  triangle. 

22.   5  and  6  are  11. 


ANSWERS   AND  SOLUTIONS 


207 


23.  A,  $25.63;  B,  $19.15;  C,  $15.32. 

24.  They  borrowed  one  sheep,  which  made  18.     After  divid- 
ing they  had  one  left,  which  was  returned  to  the  owner. 

25.  Only  6  cats. 
26. 


1 

Wi 

fe'8 

Pjirt 

- 

3 

4 

28.  The  pickets,  standing  vertically,  are  supposed  to  be  uni- 
formly the  same  distance  apart  at  the  base ;  practically  there 
would  be  the  same  number  as  at  the  top  of  the  elevation,  if 
these  pickets  were  extended  downward  to  a  common  level. 

29.  8  cats. 

30.  In  each  case  the  middle  digit  is  9  and  the  digit  before  it 
(if  any)  is  equal  to  the  difference  between  9  and  the  last  digit. 

31.  Subtract  14  from  the  result  given,  and  obtain  a  number 
of  two  digits  which  are  the  numbers  originally  chosen.  The 
digit  in  tens*  place  is  the  number  that  was  multiplied  by  5. 

33.  If  the  second  remainder  is  less  than  the  first,  the  figure 
erased  is  the  difference  between  the  remainders ;  but  if  the 
second  remainder  is  greater  than  the  first,  the  figure  erased 
equals  9,  minus  the  difference  of  the  remainders. 

.  34.    I  make  my  additions  so  that  the  sums  are  respectively, 
12,  23,  34,  45,  56,  67,  78,  and  89. 

36.    $8and  the  boots. 

36.  Take  the  goose  over,  return  and  take  the  corn  over, 
bring  the  goose  back,  take  the  fox  over,  then  return  for  the 
goose. 


208 


MATHEMATICAL   WRINKLES 


37.  Gravity  would  cause  the  ball  to  descend  toward  the 
center  of  the  earth  with  an  increased  velocity,  but  coming  con- 
stantly to  a  point  of  less  motion  in  the  earth,  it  would  soon 
scrape  on  the  east  side  of  the  hole,  until  it  passed  the  center, 
where  it  would  be  constantly  passing  points  having  a  faster 
motion  than  the  center;  it  would  soon  scrape  on  the  opposite 
side ;  the  friction  thus  retarding  the  motion,  it  would  pass  and 
repass  the  center  of  the  earth  until  it  would  finally  come  to  rest 
at  this  point. 


From  "  Curious  Cobwebs." 


38. 


39.   With  a  one,  a  three,  a  nine,  and  a  twenty-seven  pound 
weight. 
40. 


41.  Sometimes,      Mfcooorf^oro^occDo. 

42.  If  so,  by  the  same  logic,  you  can  multiply  eggs  by  eggs 
and  get  square  eggs,  or  multiply  circles  by  circles  and  get 
square  circles.  It  is  impossible  to  multiply  feet  by  feet  for 
the  principles  of  multiplication  are  —  (1)  The  ihultiplier  must 
be  regarded  as  an  abstract  number.  (2)  The  multiplicand  and 
product  must  be  like  numbers. 

43.  No. 


ANSWERS  AND  SOLUTIONS  209 

45.  I  will  always  have  mouey. 

46.  (i)  B  pushes  P  into  A.  (ii)  E  returns,  pushes  Q  up  to 
P  in  Af  couples  Q  to  P,  draws  them  both  out  to  F,  and  then 
pushes  them  toE.  (ii  i )  P  is  now  uncoupled,  the  engine  takes  Q 
back  to  Af  and  leaves  it  there,  (iv)  The  engine  returns  to  P, 
pulls  B  back  to  C,  and  leaves  it  there,  (v)  The  engine  run- 
ning successively  through  F,  D,  and  B,  comes  to  A^  draws  Q 
out,  and  leaves  it  at  B. 

—  From  Ball's  "  Mathematical  Recreations." 

47.  Place  5  on  4,  2  on  1,  11  on  10,  and  8  on  7. 

48.  Fill  the  3-gallon  cask  and  pour  it  into  the  5.  Fill  it 
again  and  pour  into  the  5  until  the  5  is  full.  There  is  now  1 
gallon  left  in  the  3.  Pour  back  the  5  into  the  8,  and  the  one 
gallon  left  in  the  3  into  the  5.  Then  fill  the  3  and  pour  into 
the  5,  making  4  gallons  in  the  5-gallon  cask,  or  one  half  of 
the  8  gallons. 

49.  $20. 

63.   80.69 -f  .74  4- .5.  54.   fJI  +  f^. 

55.  78 +  15  + \/9  + ^el- 
se.  1x2x3x4x5x6x7x8x9x0=0. 

67.    gg-i-\/4x9  .  18 
70^— 6~ 

58.  Suppose  he  starts  from  F.  Then  he  may  take  either  of 
two  routes. 

(1)  FBAUTPO  NC  DE  JKLMQRS  HGF. 

(2)  FBAUTS  RKLMQPO  NC  DEJHGF. 

Rule.  The  route  from  any  town  may  he  found  by  either  of  the 
folloicing  ruleSy  in  which  r  denotes  he  is  to  take  the  road  to  the 
right,  and  I  denotes  that  he  is  to  take  the  road  to  the  left. 

(1)  rrrlllrlrlrrrlllrlrl. 

(2)  lllrrrlrlrlllrrrlrlr. 

59.  Second  class.  The  hand  is  the  P,  the  boat  the  TT,  and 
the  water  the  F. 


210 


MATHEMATICAL   WRINKLES 


60.    $.75     $75.      63.   28  eggs.     64.   32+ feet.      65.    $2.50. 

66.  They  sell  49,  28,  and  7  at  the  rate  of  7  for  a  cent ;  then 
1,  2,  and  3  at  3  cents  each ;  hence  each  one  receives  10  cents. 

67.  39.79+  pounds.  69.    See  No.  48. 

70.  Answer,  21.  It  cannot  be  greater  than  the  smallest 
number,  27 ;  it  cannot  be  27,  since  the  remainders  would  then 
be  different.  By  dividing  these  numbers,  one  by  another,  48 
by  27,  90  by  48,  174  by  90,  we  find  the  remainders  to  be  21,  42, 
84,  the  last  two  being  multiples  of  the  first.  Now  dividing 
the  numbers  by  these  remainders  (21  and  the  multiples),  27  by 
21,  48  by  42,  90  by  84,  and  174  by  168,  the  next  multiple  of 
21,  we  obtain  a  remainder  which  is  the  same  in  each  case ;  we 
therefore  conclude  that  dividing  all  the  numbers  by  21  would 
give  a  like  result. 

71  10  feet 


5  ft. 

5  ft. 

•=■ 

•r 

72.    |.  73.    XIX.     Take  the  1  away  and  have  XX. 

74.  1^  pounds. 

75.  Invert  the  6  to  make  it  9.     The  whole  number  may  be 
either  918  or  198. 

76.  1.25.  /     '\, 

77.  Tree./'  ^v  .Tree 


Tree  i 


Land 
20  acres 


•  Tree 


ANSWERS  AND  SOLUTIONS  211 

78.  The  method  would  have  been  incorrect.  The  division 
would  have  been  in  favor  of  the  tenant.  The  landlord  would 
have  received  J  of  45  bushels  when  he  was  entitled  to  I  of  the 
45  bushels  and  also  to  J  of  the  18  bushels.  In  other  words,  he 
should  have  received  |  of  (45  + 18),  or  25^  bushels.  The  land- 
lord would  have  lost  the  difference  between  25^  bushels  and 
18  bushels,  or  7|  bushels. 

79.  792.  80.    20.  81.    0. 

82.  By  immersing  them  in  a  vessel  of  water. 

83.  B  hoes  six  the  most.  84.   3.  85.    3'  -}-  3. 

86.  IX  =  9.     Cross  the  I  and  make  it  XX. 

87.  21  days,  because  two  of  the  ears  are  his  own. 

88.  43  days.  89.    64 
90. 


5  1  io!ie« 

! 

',  )nche« 

^fr. 

91.  First,  the  two  sons  cross,  then  one  returns.  Second,  the 
man  crosses  and  the  other  son  returns.  Third,  both  sons  cross, 
and  then  one  returns.  Fourth,  the  lady  crosses  and  the  other 
son  returns.     Fifth,  the  two  sons  cross. 

92.  199.  93.    Infinity.  94.    2§. 

95.   Tf  96.   38  -  3,  or  22  +  2.  97.    29  days. 

98.   Ans,   987,654,321. 

When  multiplied  by  18  =  17,777,777,778. 
When  multiplied  by  27  =  26,666,666,667. 
When  multiplied  by  36  =  35,555,565,556. 
When  multiplied  by  45  =  44,444,444,445. 
When  multiplied  by  54  =  53,333,333,334. 
WJien  multiplied  by  63  =  62,222,222,223. 
When  multiplied  by  72  =  71,111,111,112. 
When  multiplied  by  81  =  80,000,000,001. 


212  MATHEMATICAL  WEIXKLES 

When  multiplied  by  99  =  97,777,777,779. 
When  multiplied  by  9  =  8,888,888,889. 
When  multiplied  by  90  =  88,888,888,890. 

The  same  is  true  of  higher  multiples  of  nine.     Thus, 

108  X  987,654,321  =  106,666,666,668. 

117  X  987,654,321  =  115,555,555,557. 

99.  Three  cents  for  each  seven  and  nine  cents  each  for  the 
remainder. 

100.  It  will  make  no  difference  as  long  as  he  jumps  on  the 
deck.  Should  he  jump  off  the  boat,  then  the  effect  would  be 
different. 

101.  10.  102.    33. 

103.  When  the  figures  added  make  nine  or  some  multiple 
of  nine. 

104.  3  +  0  +  2  +  0-1-1+1=7.  9-7  =  2.  Two  is  wanting 
to  make  a  multiple  of  nine,  therefore  2  placed  anywhere  in,  or 
before,  or  after  the  number,  will  make  it  divisible  by  nine. 

105.  11  grooms  and  15  horses.  106.    11  cents. 
107.   301.                             108.    300  pounds;  also  300  pounds. 

109.  Because  muscles  and  bones  are  heavier  than  fat.  The 
specific  gravity  of  a  fat  man  is  therefore  less  than  that  of  a 
lean  one. 

110.  The  ball  which  is  thrown  has  time  to  impart  its  motion 
to  the  board ;  but  the  one  fired  has  not. 

111.  Move  from  1  to  6,  4  to  1,  7  to  4,  2  to  7,  5  to  2,  8  to  5, 
and  3  to  8. 

112.  8888,  when  halved  equals  0000. 

113.  4±2V3. 

114.  May  have  any  shape  ;  square. 


ANSWERS   AND   SOLUTIONS 


213 


115.   1  is  the  only  integer.     This  is  however  true  of  any 
number  between  0  and  2. 


116. 

2  +  2  =  2^,  orO  +  0  = 

0\ 

117. 

142,857. 

118. 

12. 

121. 

64 

16 

56 

96i 

25 

89  +  1+3  +  7  =  100. 

36 

47 

98 

2 

100 

8 

4 

3 

71 

29 

100 

100 

Many  other  solutions. 


122. 


G 

10 

3 

15 

11 

7 

14 

2 

16 

4 

9 

5 

1 

13 

8 

12 

123. 


124. 


125.  Arrange  the  figures  as  in  (12)  except  use  such  figures 
to  place  opposite  each  other  that  when  added  make  20.  Use 
10  at  center. 


126. 


127. 


8 

1 

6 

8 

6 

T 

4 

9 

2 

129.    7. 

132.    B,  C,  and  A  respectively. 


214  MATHEMATICAL   WRINKLES 

134.  (1)   320432  =  1026753849. 

(2)  990662  =  9814072356. 

(3)  (4)  The  least  solutions  which  have  been  found  of  (3),  (4) 
are  identical : 

101010101010101012  =  10203  ...  080908  ...  30201,  but  there 
are  probably  lower  numbers  suitable.  If  numbers  beginning 
with  zero  were  admissible,  then  much  lower  numbers  would 
suffice,  e.  g.,  01111111110^  =  001234567898765432100. 

— From  "Mathematical  Reprint." 

135.  25.3  rods. 

136.  Let  a;  =  A's  ability  and?/  =  B's  ability.  Since  A  can 
dig  the  ditch  in  the  same  time  that  B  shovels  the  dirt,  x:y  = 
the  labor  required  to  dig  it  :  the  labor  required  to  shovel  the 
dirt. 

And  since  B  can  dig  twice  as  fast  as  A  can  shovel,  ^y:x  —  the 
labor  required  to  dig  it  :  the  labor  required  to  shovel  the  dirt. 
.-.  x:y  =  ^y:x. 
.\y  =  xV2  =  lAUx. 

A  should  receive  -^—  of  $10,  or of  $10  =  —— 

x-\-y  a; +  1.414  a;  2.414 

of  $  10  =  $  4.14.     B  received  $  5.86. 

138.  This  is  a  mere  trick.  When  trains  meet  they  must  be 
at  the  same  distance  from  a  given  point. 

139.  0  is  the  only  possible  digit  to  satisfy  the  first  addend 
total.  2  being  "  carried  "  to  the  second  column,  2  is  found  to  be 
the  next  missing  number.  The  third  column  total  must  be 
either  23  or  33.  On  trial,  the  former  is  found  to  be  wrong,  and 
the  only  two  numbers  making  18,  so  as  to  give  33  as  total  of 
third  line,  are  9  and  9.  Proceeding,  we  find  that  9  is  wanted  in 
the  fourth  line,  and  in  the  sixth  line  two  O's.  Only  these 
numbers  will  satisfy,  and  the  answer  is  proved  by  adding. 

140.  The  only  number  to  satisfy  the  product  by  5  is  3,  and 
this  being  supplied  the  remaining  numbers  are  easily  found. 


ANSWERS   AND  SOLUTIONS  215 

The  only  possible  numbers,  in  the  multiplier  are  3  and  8 ;  of 
these  we  see  that  8  is  the  one  required.  The  third  missing 
number  is,  of  course,  6. 

141.    If  the  question  be  put  down  in  skeleton,  we  shall  be 
more  readily  able  to  supply  the  missing  links. 

)529566(*** 


XX 


2466 


2225 
"642" 


2244 

1683 


The  last  remainder  being  542,  we  see  that  2225  —  542,  i.e. 
1683,  must  be  the  last  multiple  of  the  divisor.  Similarly, 
2466  -  222  leaves  2244  as  another  multiple.  The  G.  C.  M. 
of  2244  and  1683  is  561,  and  this  number  is  greater  than  the 
largest  remainder  (542),  hence  561  is  the  divisor.  The  quo- 
tient, by  division,  will  be  found  to  be  943. 

142.  The  middle  digit  remains  unaltered,  and  since  in  add- 
ing the  second  digit  is  7  (1  being  carried),  the  second  line  total 
must  be  17,  and  therefore  8  must  be  the  middle  digit.  Again 
the  first  digit  total  must  be  10,  and  the  only  possible  addends 
are  9  and  1,  8  and  2,  7  and  3,  6  and  4,  5  and  5.  By  testing  it 
will  be  seen  that  9  and  1  are  the  only  two  numbers  fulfilling 
conditions.     Ans.  981  and  189. 

143.  Instead  of  multiplying  by  409,  she  actually  multiplied 
by  49,  therefore  her  answer  was  short  of  the  true  answer  by 
(409  —  49),  i.e.  360,  times  the  multiplicand,  and  we  are  told  this 
was  328,320 ;  328,320  -«-  360  =  912,  the  multiplicand. 

—  From  "  Arithmetical  Wrinkles.** 

144.  3  ft. 

145.  (1)  Webster^s  Dictionary  says  that  after  the  sign  I3 
for  D,  the  character  3  (called  the  apostrophus)  was  repeated 


216  MATHEMATICAL   WRINKLES 

one  or  more  times,  each  repetition  having  the  effect  of  multi- 
plying Iq  by  10.  To  represent  a  number  twice  as  great,  C 
was  repeated  as  many  times  before  the  stroke,  I,  as  the  3  was 
given  after  it.     Hence,  one  billion  is  written, 

CCCCCCCioooOOOO 
(2)  A  bar  placed  over  any  number  in  the  Eoman  notation 
multiplies  the  original  value  by  1000 ;  hence  two  bars  placed 
over  it  would  be  a  thousand  times  a  thousand  times  its  initial 
value,  and  M  =  1000  x  1000  x  1000  =  1,000,000,000. 

— From  "  The  School  Visitor." 

146.  Let  AC  locate   the   ditch  and  0  the   point   required. 

The  triangles  AOD  and  COB 
have  equal  altitudes,  DG  and 
BF',  hence,  their  bases  AO 
and  DC  must  be  to  each  other 
as  their  areas,  or  as  2  to  3, 

and  0  may  be  -f  of  the  distance  from  ^  to  (7  or  from  C  to  A. 

147.  Let  X,  y,  and  z  be  the  three  digits.  Then,  (100x-\-10y 
-\-z)-  (100 ;?  4- 10  2/  +  a^)  =  99  a?  -  99  2.  Consequently,  the  dif- 
ference for  any  set  of  three  digits  is  9  x  11  (a?  —  z). 

The  result  is  always  99  times  the  difference  of  the  extreme 
digits. 

148.  99||-. 

149.  Multiply  the  selected  number  by  nine,  and  use  the  prod- 
uct as  the  multiplier  for  the  larger  number.  It  will  be  found 
that  the  result  is  in  each  case  the  "  lucky  "  number,  nine  times 
repeated. 

150.  The  father  was  three  times  the  age  of  his  son  15| 
years  earlier,  being  then  6b\  while  his  son  was  18|-.  The  son 
will  have  reached  half  his  father's  age  in  3  years'  time,  being 
then  37,  while  his  father  will  be  74. 

151.  45. 


ANSWERS  AND   SOLUTIONS  217 

152.  The  asterisks  indicate  the  figures  to  be  expunged. 

*11 

33* 

«** 

77* 
*■** 

iTTi 

153.  By  the  conditions  there  were  twelve  children  in  all  and 
each  has  now  nine,  then  each  parent  had  three  children  when 
married,  making  six  arrivals  within  ten  years. 


156.                                    50^ 

80H 

49f* 

19| 

100 

100 

Many  other  solutions. 

157. 

1  =  44-^44. 

16  =  4x4-4-h4. 

2  =  4^4  +  4-j-4. 

17  =  4x4-1-4-- 4. 

3=(4-|-44-4)-i-4. 

18  =  4h-.4  +  4-|-4. 

4  =  V(4x4x4-j-4). 

19  =  (4 -i- 4 -.4) -J- .4. 

6  =  V(4x4)+4-^4. 

20  =  4h- .4  +  4^.4. 

6  =  4^-.4-V(4x4). 

21  =  (4.4  +  4) +  .4. 

7  =  44-1-4-4. 

22  =  (4  +  4)-.4+V4. 

8  =  (4-f4)x  (4-5-4). 

23  =  4h-(.4x  .4)- V4. 

9  =  4-h4-h4-^-4. 

24  =  (4  +  4)-.4+4. 

10  =  4--.4-f  4-4. 

25  =  (4  +  4+V4)--.4 

11  =4-- .4-1-4-!- 4. 

26  =  4x4  +  4-*- .4. 

12  =  4x4-V(4x4. 

27  =  4--(.4x.4)+V4. 

13  =  44 -i-4-hV4. 

28  =  44-4x4. 

14  =  4^.4  + V(4x4). 

29  =  4 -J- (.4  X. 4) +4. 

15  =  44-^-4  +  4. 

30=(4  +  4  +  4)-i-.4. 

158.   The  figure  that  occurs  in  the  quotient  is  the  diiference 
between  the  first  and  last  figures  of  the  number  taken. 


218  MATHEMATICAL   WRINKLES 

159.  The  figure  erased  is  the  first  remainder  minus  the 
second,  or  if  the  first  is  not  greater  than  the  second,  then  it  is 
the  first  +  9  —  the  second. 

—  2      4-4 

160.  Yes.     For  example  =  — — . 

+  5      - 10 

161.  Every  even  number  contains  2  as  a  factor  and  e very- 
alternate  even  ntimber  contains  4  as  a  factor ;  hence,  the  prod- 
uct of  any  two  consecutive  even  numbers  is  divisible  by  8. 

162.  Indeterminate. 

163.  A  gets  seven  ninths  as  much  as  B  per  rod ;  hence  to 
get  equal  money  A  must  dig  nine  sevenths  as  much.  Dividing 
100  rods  in  proportion  of  9  to  7,  A  must  dig  56.25  rods  and  B 
43.75  rods.  For  actual  work  each  gets  thus  an  equal  sum, 
^98.4375,  and  we  may  now  infer  that  the  balance  of  the  money 
should  be  equally  divided,  giving  each  $  100. 

164.  3  ounces.  165.,  10  cents. 

166.  There  is  no  change  in  the  weight,  since  the  weight  of 
the  fish  is  the  same  as  the  weight  of  the  water  displaced. 

167.  A  rectangular  tank  twice  as  wide  as  it  is  deep  with  a 
square  base. 

168.  $13.75. 

169.  300. 

170.  The  bird  is  heavier  than  the  air  and  supports  itself  by 
striking  down  upon  the  air.  The  increase  in  weight  caused  by 
these  strokes  would  undoubtedly  be  the  difference  between  the 
weight  of  the  bird  and  the  weight  of  its  displacement  of  air. 

171.  Suppose  John's  rate^of  work  is  w  times  James'. 
Then,  by  the  first  condition, 

James'  work  :  John's  work  : :  3  :  w, 
and  by  the  second  condition, 

James'  work  :  John's  work  ::w:l. 
.*.  3  :  w::w:l. 


ANSWERS   AND   SOLUTIONS  219 

Then  w  =  V3,  a  mean  proportional. 

Then  $10  must  be  divided  between  John  and  James  in  the 
ratio  of  1 :  V3,  which  makes  John's  share  $3.66,  and  James' 
share  S6.34. 

173.  5. 

174.  Subtract  from  the  higher  multiple  of  9. 

176.  A  mile  square  can  be  no  other  shape  than  square;  the 
expression  names  a  surface  of  a  certain  specific  size  and  shape. 
A  square  mile  may  be  of  any  shape ;  the  expression  names  a 
unit  of  area,  but  does  not  prescribe  any  particular  shape. 

179.   36  cents. 


180. 

2. 

181. 

SIX 

IX 

XL 

IX 

X 

L 

s  . 

I 

X 

182. 

41.78+  feet. 

183.   The  correct  answer  is  \.  . 

185.  There  are  several  solutions. 
The  vessels  can  hold 
Their  contents  to  begin  with  are 
First,  make  their  contents 
Second,  make  their  contents 
Third,  make  their  contents    • 
Fourth,  make  their  contents 
Fifth,  make  their  contents 
Last,  make  their  contents 

186.  He  lost. 

OJA  Let  -  be  the  required  fraction. 

187.  — .  y 

^^  Then  ?  pounds  +  -  shillinp  +  -  pence  =  1  pound. 


One  is  as 

follows : 

24  oz.    13  ( 

oz. 

11  oz. 

5  oz. 

24           0 

0 

0 

0           8 

11 

5 

16           8 

0 

0 

16           0 

8 

0 

3         13 

8 

0 

3           8 

8 

6 

8           8 

8 

0 

220  MATHEMATICAL  WRINKLES 


Reducing  all  to  pence,  we  have, 

240  X  +  12  a;  +  X 


=  240. 


.•.253^  =  240. 

y 

Solving,  ^  =  ?^. 
""  y     253 

188.  The  number  45  is  the  sum  of  the  digits  1,  2,  3,  4,  5,  6, 
7,  8,  9.  The  puzzle  is  solved  by  arranging  these  in  reverse 
order,  and  subtracting  the  original  series  from  them,  when  the 
remainder  will  be  found  to  consist  of  the  same  digits  in  a  dif- 
ferent order,  and  therefore  making  the  same  total. 

987654321  =  45 

Thus,  123456789  =  45 

864197532  =  45 

191.  One  traveled  ten  times  around  the  world,  and  the 
other  remained  at  home,  or  both  traveled  around  the  earth 
in  opposite  directions,  the  sum  of  the  two  sets  of  circumnavi- 
gation amounting  to  ten. 

192.  (a)  They  start  in  a  high  latitude  and  on  the  same 
meridian,  both  going  east  or  west,  (h)  They  start  in  a  high 
latitude,  both  on  the  same  parallel  and  travel  south  (or  if  in  a 
high  southern  latitude  they  would  travel  north),  (c)  They 
may  start  each  ten  miles  from  the  north  pole  180°  of  longitude 
apart,  and  each  travels  five  miles  south. 

193.  They  travel  from  the  North  Pole  to  the  South  Pole,  or 
from  the  South  Pole  to  the  North  Pole. 

194.  Standing  on  the  North  Pole. 

195.  It  will  never  come  up. 

196.  40  pounds. 

197.  (9|)^  9||;  §^;  -^9^  +  f 

200.  All  that  is  necessary  is  to  deduct  25  from  the  sum 
named.  This  will  give  a  remainder  of  two  figures,  represent- 
ing the  points  of  the  two  dice. 


ANSWERS   AND   SOLUTIONS 


221 


201.  Subtract  250  from  the  sum  named.  This  will  give  a 
remainder  of  three  figures,  representing  the  points  of  the  three 
dice. 

202.  I  Solution.  Sam  and  John  each  take  2  full  casks,  2 
empty  casks,  and  3  half-full  casks ;  and  James,  3  full  casks, 
3  empty  casks,  and  1  half-full  cask. 

II  Solution.  Sam  and  John  each  take  3  full  casks,  1  half- 
full  cask,  and  three  empty  casks ;  and  James  1  full  cask,  5 
half-filled  casks,  and  1  empty  cask. 

204.  This  problem  is  susceptible  of  various  answers,  equally 
correct,  according  to  the  value  assigned  to  the  smallest  part, 
or  unit  of  measurement.  If  this  unit  of  measurement  be  1,  the 
number  will  be  1 -f  40 -f- 400  +  500  =  941.  If  the  unit  be  2, 
the  number  will  be  2  +  80 -|- 800 -f  1000  =  1882,  and  so  on 
ad  infinitum, 

205.  2619. 

206.  Find  the  center  of  either  side  of  a  given  square,  and 
cut  the  card  in  a  straight  line  from  that  point  to  one  of  the 
opposite  corners,  as  shown  in  the  small  figure.  Treat  four  of 
the  five  squares  in  this  manner.     Rearrange  the  eight  segments 


thus  made  with  the  uncut  square  in  the  center,  as  shown  in 
the  larger  figure,  and  you  will  have  a  single  perfect  square. 

—  From  "Mechanical  Puzzles." 


222 
207. 


MATHEMATICAL  WRINKLES 


15    inches 


3  in. 

yy'^Q>  in. 

^^Q,  in. 

208.    Cut   as   indicated   in  first  figure,  and  rearrange  the 
pieces  as  shown  in  the  second  figure. 


209 


9  ..8  >  2  "  JO 

210.  Count  and  mark  every  ninth  one,  marking  it  "Turk" 
until  15  are  marked.     Mark  the  remaining  ones,  "  Christians." 

211.  16|.        213.   7  and  5.        214.    10|  hours.         217.    60. 
215.    72.  218.    28.  216.   PRECAUTION. 

219.    Divide  tlie  cross  as  indicated  in  the  first  figure  and 
rearrange  as  shown  in  the  latter. 


\ 
\ 

\  rz 7 Ti 

I -\ 1  ^"^^'-^         '' 


ANSWERS  AND   SOLUTIONS 


223 


224.  Subtract  the  smallest  number  from  each  of  the  others. 
The  G. CD.  of  the  differences  is  the  required  divisor.  An- 
swer, 2. 

226.  The  number  of  shoes  equals  the  number  of  persons. 

227.  27.3083+  inches. 

228.  There  would  be  no  difference  in  the  weight  when  the 
bird  perched  or  flew.  The  air  which  supports  the  bird  rests 
on  the  bottom  of  the  cage.  If  the  same  cage  had  no  top,  the 
same  would  hold.  If  it  had  no  bottom,  there  would  still  be  no 
difference  in  weight.  In  this  case  the  flight  of  the  bird  would 
tend  to  produce  a  vacuum  just  under  the  top,  and  the  air  above 
the  cage  would  press  downward  with  a  force  equal  to  the 
weight  of  the  bird.  If  both  top  and  bottom  were  removed, 
there  would  be  a  difference  equal  to  the  weight  of  the  bird. 

—  From  "  School  Science  and  Mathematics." 


232. 


rri 

w$ 

^■" 

^m 

■^ 

^^^ 

^P 

losm^ 

t 

y 

? 

/, 

A 

/ 

aq- 

7 

/ 

/ 

/ 

# 

/ 

A 

V 

/ 

/ 

/ 

/ 

:  v. 

i. , 

/ 

/ 

/ 

^ 

-^ 

V 

/ 

z 

/ 



1      1 



233.  2||U|.  inches  and  J^ff  inches. 
(2itHf)'  +  (ttftt)'  =  17. 

234.  Similar  solids  are  to  each  other  as  the  cubes  of  cor- 
responding lengths.    Therefore  the  volumes  of  the  balls  are  to 


224 


MATHEMATICAL   WKINKLES 


each  other  as  1^  is  to  2^  or  as  1  to  8.  By  adding  1  to  8  we  get 
9.  9  is  the  sum  of  these  two  perfect  cubes.  We  must  now 
find  two  other  numbers  whose  cubes  added  together  make  9. 
These  numbers  must  be  fractionah     They  are  lifffrg-fl-lf-J 

fppt  and    676702467503    fppj- 
iet!b  ana   3  4  8  6TT6^8T6"6"0   ^^^^• 

235.    Yes.     By  2,071,723  and  5,363,222,357. 


T — I — r 


^-  +  + 
+  +[+ 


J I L 


236.  I  entered  the  room  C  because  I  put  my  foot  and  part 
of  my  body  in  it,  and  I  did  not  enter  the  other  room  twice, 
because  after  once  going  in  I  never  left  it  until  I  made  my 
exit  at  B.     This  is  the  only  possible  solution. 

237. 


AI^SWERS  AND   SOLUTIONS 


225 


238.  Bisect  AB  at  D  and  BC  at  E ;  produce  AE  to  F  making 
EF  equal  to  EB-,  bisect  AF  at  O 
and  describe  the  arc  AHF ;  produce 
£5  to  H,  and  J&fT  is  tlie  length  of 
the  side  of  the  required  square; 
from  E  with  distance  EIIj  describe 
the  arc  JIJ  and  make  JK  equal  to 
BE ;  now  from  the  points  D  and  K 
drop  perpendiculars  on  EJ  at  L  and 
3f.  If  you  have  done  this  accu- 
rately, you  will  now  have  the  required  directions  for  the  cuts. 

241.  24.  Keduce.  the  length  of  the  block  by  half  an  inch. 
The  small  block  constitutes  the  waste.  Cut  the  other  piece 
into  three  pieces  each  IJ  inches  thick.  Each  of  these  may  then 
be  cut  into  eight  blocks. 

242.  There  are  eleven  times  in  twelve  hours  when  the  hour 
hand  is  exactly  twenty  minute  spaces  ahead  of  the  minute 
hand.  If  we  start  at  four  o'clock  and  keep  on  adding  1  hour 
5  minutes  27^  seconds,  we  shall  get  all  these  eleven  times, 
the  last  being  2  hours,  54  minutes,  32^  seconds  past  twelve. 
Another  addition  brings  us  back  to  four  o'clock,  but  at  this 
time  the  second  hand  is  nearly  twenty-two  minute  spaces  be- 
hind the  minute  hand,  and  if  we  examine  all  our  eleven  times, 
we  shall  find  that  only  in  one  case  is  the  second  hand  the 
required  distance.  This  time  is  54  minutes,  32-j^  seconds 
past  2.       ^ 

243.  A 

(6) 


226  MATHEMATICAL   WEINKLES 

244.  1  —  2  — 3  — 4  — 5;  1  —  2  —  4  —  5—3;  1  —  3  —  2  — 
5  — 4;  1—3  — 4— 2  — 5;  1—4  —  2  —  3  —  5;  1  —  4-3  — 
5  —  2. 

245.  Let  A,  B,  C,  T>,  E,  F,  and  G  represent  the  seven  men. 
The  way  of  arranging  them  is  as  follows :  — 


ABC 

D  E  F  G 

A  C  D 

B  G  E  F 

A  D  B 

C  F  G  E 

A  G  B 

F  E  C  D 

AFC 

E  G  D  B 

A  E  D 

G  F  B  C 

ACE 

B  G  F  D 

A  D  G 

C  F  E  B 

A  B  F 

D  E  G  C 

A  E  F 

D  C  G  B 

AGE 

B  D  F  C 

A  F  G 

C  B  E  D 

A  E  B 

F  C  D  G 

A  G  C 

E  D  B  F 

A  F  D 

G  B  C  E 

246.    3ift. 


247.  The  bag  contained  either  79,  160,  241,  322,  or  403,  etc. 

248.  Twenty-six  transfers  are  necessary.     Move  the  cars  so 
as  to  reach  the  following  positions  :  — 

£-567  8 


1234 
E  56 


123     87 
56 

^312     87 
E 


=  10  transfers 
=  2  transfers 
=  5  transfers 
9  transfers. 


8765432  1 

250.    If  there  were  twelve  ladies  in  all,  there  would  be  132 
kisses  among  the  ladies  alone,  leaving  twelve  more  to  be  ex- 


ANSWERS  AND   SOLUTIONS  227 

changed  with  the  curate  —  six  to  be  given  by  him  and  six  to 
be  received.  Therefore  of  the  twelve  ladies,  six  would  be  his 
sisters.  Consequently,  if  twelve  could  do  the  work  in  four 
and  a  half  months,  six  ladies  would  do  it  in  nine  months. 

252.  Only  three  revolutions  are  necessary. 

Number  the  nests  from  1  to  12  in  the  direction  the  person 
travels.  Transfer  the  egg  in  nest  No.  1  to  nest  No.  2,  in  No.  5 
to  nest  No.  8,  in  No.  9  to  No.  12,  in  No.  3  to  No.  6,  in  No.  7  to 
No.  10,  in  No.  11  to  No.  2,  and  complete  the  last  revolution  to 
nest  No.  1. 

This  can  also  be  done  by  transferring  the  egg  in  nest  No.  4, 
to  No.  7,  in  No.  8  to  No.  11,  in  No.  12  to  No.  3,  in  No.  2  to  No.  5, 
in  No.  6  to  No.  9,  in  No.  10  to  No.  1. 

253.  He  divided  the  rope  in  half.  He  simply  untwisted  the 
strands  and  divided  it  into  two  ropes,  each  being  of  the  original 
length  of  the  rope.  He  then  tied  these  two  ropes  together 
and  had  a  rope  almost  twice  as  long  as  the  original  rope. 

254.  26.0299626611 71957726998490768328505774732373764 
7323555652999. 

255.  I  reached  the  shore  with  little  difficulty.  I  fastened 
one  end  of  the  trot  line  to  the  stern  of  the  boat,  and  then  while 
standing  in  the  bow,  gave  the  line  a  series  of  violent  jerks 
thus  propelling  the  boat  forward. 

256.  C*s  age  at  A's  birth  4-  A's  present  age  =  A's  present 
age  -f  B's ;  then  C's  age  at  A*s  birth  =  B's  present  age.  By 
the  second  condition,  A's  age  —  3  =  |  (B's  4-  4),  from  which 
A's  age  =  f  B's  age  +  6  years.  The  difference  between  A's  and 
B's  present  ages  =  B's  age  at  birth  of  A.  Therefore  \  of  B's 
present  age  —  6  years  =  B's  age  at  A's  birth,  and  5^  (J  B's  age 
—  6  years)  =  -y-  of  B's  present  age,  from  which  -y-  B's  age  —  33 
years  =  B's  age,  or  88  years.  C's  age  at  A's  birth  was  also  88 ; 
B's,  88  -i-  5J,  or  16  years.  A's  present  age  is  88  — 16  =  72 
years ;  B's  88,  and  C's  88  -f  72  =  160  years. 

257.  £2,567  18  s.  9|d. 


SHORT  METHODS 

Business  men  everywhere  complain  that  the  schools  teach 
neither  accuracy  nor  rapidity  in  calculations.  They  claim  that 
the  pupils  must  learn  facts  and  principles  and  have  much 
practice  in  the  application  of  principles;  that  because  a  boy 
can  apply  a  principle  to-day  is  no  guarantee  that  he  will  have 
the  same  knowledge  and  ability  tomorrow ;  that  eternal  vigi- 
lance is  not  only  the  price  of  liberty,  but  also  the  price  of 
proficiency. 

"  The  mechanic  who  is  not  skillful  in  the  use  of  his  tools 
will  never  rise  above  poor  mediocrity;  the  pupils'  arithmetical 
tools  are  figures,  and  unless  he  can  handle  these  with  facility 
and  accuracy,  he  must  ever  remain  a  plodder,  a  waster  of  time, 
and  a  blunderer  upon  whose  results  none  can  depend." 

We  are  living  in  a  fast  age,  an  age  of  steam  and  electricity, 
when  results  are  attained  by  lightning  methods. 

ADDITION 

There  are  no  short  cuts  in  addition ;  every  figure  in  every 
column  must  be  added  to  ascertain  the  amount.  Nevertheless 
the  time  required  to  perform  an  operation  in  addition  can  be 
substantially  shortened  in  the  following  ways : 

1.  By  making  plain,  legible  figures. 

2.  By  placing  units  of  a  certain  order  immediately  beneath 
units  of  a  like  order. 

3.  By  omitting  the  "  ands  "  and  "  ares.'* 

4.  By  making  combinations  of  10. 

5.  By  double  column  adding. 


SHORT  METHODS 


229 


1.  Civil  Service  Method 

When  long  columns  are  to  be  added,  the  following  method 
will  be  found  practical. 

485 
576 
324 
449 
625 
264 
33 
29 
24 


To.  insure  accuracy,  add  each  column 
from  top  downwards  as  well  as  from  bot- 
tom upwards. 


2723 


OPBBATION 


24 


21 

83 
62 
63 
49 


Two-Column  Adding 


Explanation.  To  add  2  columns  at  a  time,  begin  with 
the  number  at  the  bottom  and  add  the  units  of  the  number 
next  above,  and  then  add  the  tens,  naming  the  totals  only. 
Continue  in  this  way  until  all  the  numbers  are  added. 
Thus,  the  given  example  would  read  49,  62,  102,  104,  164, 
167,  247,  248,  268,  274,  304,  808,  328. 


3. 

OPERATION 

142 
881 
212 
468 
1203 


To  add  Tliree  or  More  Columns 


Explanation.  Three  columns  or  more  may  be  added  at 
one  time  by  extending  the  two-column  method  to  include 
all  the  columns  desired.  Thus,  468,  470, 480,  680,  681,  761, 
1061,  1063,  1103,  1203. 


4.  A  Jap  Method  of  Adding 

Illustrative  Example.  —  Find  the  sum  of  382,  498,  364,  899, 
842,  and  789. 


230 


MATHEMATICAL   WRINKLES 


OPERATION  Explanation.  —  Write  382   as   read.     To  add  498, 

say  4  +  3  =  7  ;  9  +  8  =  17,  write  .7,  the  dot  (.)  shows 
that  1  ten  is  to  be  carried  ;  8  +  2  =  .0. 

To  add  364,  say  3  +  7.  =  11,  the  dot  following  the 
7  increases  its  value  1  and  is  read  8. 

Continuing  this  method,  we  obtain  2.6.6.4  as  the 
result,  which  would  be  read  3774. 


382 
7.7.0 

1  1.4  4 

2  0.3.3 

2985 
2.6.6.4    Ans. 

Note.  —  To  be  an  adder  of  any  consequence,  one  ought  to  be  able  to 
add  at  least  one  hundred  figures  per  minute. 


SUBTRACTION 

There  are  three  common  methods  of  subtraction.  In  the 
following  example,  we  may  say, 

(1)  6  from  15,  9 ;  2  from  3,  1 ;  4  from  13,  9 ;  1345 

(2)  6  from  15,  9 ;  3  from  4,  1 ;  4  from  13,  9  ;  426 

(3)  6  and  9,  15  ;  2  and  1  and  1,  4 ;  4  and  9,  13.  919 

Each  of  these  methods  is  easily  understood.  The  first  is 
the  simplest  of  explanation,  and  hence  it  is  generally  taught 
to  children.  The  second  is  slightly  more  rapid  than  the  first. 
But  the  third,  familiar  to  all  as  the  common  method  of  "mak- 
ing change,"  is  so  much  more  rapid  than  either  of  the  others 
that  it  is  recommended  to  all  computers.  This  method  is 
called  the  "  Addition  Method." 


MULTIPLICATION 


The  squares  of  all  numbers  up  to  30  should  be  memorized. 
They  become  the  basis  of  further  knowledge  of  numbers.    Thus  : 


13  X  13  =  169 

14  X  14  =3  196 

15  X  15  =  225 

16  X  16  =  256 
17x17  =  289 
18  X  18  =  324 


19  X  19  =  361 

20  X  20  =  400 

21  X  21  =  441 

22  X  22  =  484 

23  X  23  =  529 

24  X  24  =  576 


25  X  25  =  625 

26  X  26  =  676 

27  X  27  =  729 

28  X  28  =  784 

29  X  29  =  841 

30  X  30  =  900 


SHORT   METHODS  231 

1. 

When  the  Multiplicand  and  Multiplier  are  Alter- 
nating Numbers 

Alternating  numbers  are  those  having  in  their  regular  order 
a  number  between  them;  as  7  and  9;  19  and  21 ;  32  and  34. 
Rule.  —  Write  the  square  of  the  intermediate  number  less  one. 

Example.— 15  x  17  =  16^  -  1  =  256  -  1  =  255. 
17  X  19  =  182  -  1  =  324  -  1  =  323. 
39  X  41  =  40»--l  =  1600 - 1  =  1599. 

Note. — The  product  of  two  numbers  having  three  intermediate  num- 
bers between  them  is  equal  to  the  square  of  the  central  number  less  4. 
Thus  9  X  13  =  112  -  4  =  117. 

2.   When  the  multiplier  is  a  composite  number. 
Multiply  328  by  42. 

OPERATION 

328 

7  Explanation.  —  The  factors  of  42  are  7  and  6.     We 


2296  multiply  328  by  7,  and  this  result  by  6  and  obtain  13,776. 

6 


13776    Ans. 

3.  When  the  right-hand  figure  of  the  multiplier  is  1. 
Multiply  23,425  by  41. 

OPERATION  Explanation.  —  Multiply  the  units'  figure  of  the  mul- 

23426  by  41       tiplicand  by  the  tens'  figure  of  the  multiplier  and  se^  the 
93700  figure  of  the  product  obtained  one  place  to  the  left  of 

060425  Ans.  units'  figure  of  the  multiplicand.  Continue  in  this  man- 
ner until  all  the  figures  of  the  multiplicand  have  been 
multiplied  by  the  figures  of  the  multiplier,  and  add  the  product,  or 
products,  thus  found  to  the  multiplicand  and  the  result  will  be  the 
product  desired. 

4.  When  the  multiplier  is  a  unit  of  any  order. 

Rule.  —  Annex  as  many  ciphers  to  the  multiplicand  as  there 
are  ciphers  in  the  multiplier. 
Thus  42  X  10  =  420 ;  21  X  100  =  2100,  etc. 


232  MATHEMATICAL  WKINKLES 

5.   When  the  multiplier  is  11. 

Rule.  —  Beginning  with  units,  add  each  term  of  the  multipli- 
cand to  the  one  preceding,  carrying  as  in  the  regular  rule. 
Multiply  1328  by  11. 

OPERATION  Explanation.— 0  +8  =  8  and  we  write  8  for  the  units' 

1328  figure  of  the  product ;  8  +  2  =  10,  we  write  0  for  tens' 

11  place  ;  2  +  3  =  5  and  1  carried  =  6  ;  we  write  6  ;  3  +  1=4, 


14608     Ans.      we  write  4  ;  1+0=1,  we  write  1  and  the  product  is  14,608. 

6.    When  the  multiplier  is  9,  99,  or  any  number  of  9's. 

Rule.  —  Annex  to  the  multiplicand  as  many  ciphers  as  the  mul- 
tiplier contains  O's,  and  subtract  the  multiplicand  from  the  result. 

Thus  43561  x  999  =  43,561,000  -  43,561  =  43,517,439,  Ans. 

7. "  To  multiply  any  two  figures  by  11. 

Rule.  —  Add  the  figures  and  place  the  result  between  them. 

Thus  42  X  11  =  462,  29  X  11  =  319,  etc. 

8.   To  multiply  by  any  number  which  ends  with  9. 
Multiply  327  by  39. 

OPERATION  Explanation.  —  The  next  number  higher  than  39  is 

327  40.     Multiplying  the  multiplicand  by  40  produces  a  re- 

40  suit  of  13,080.     The  real  multiplier  is  one  less  than  40, 

13080  therefore  by  subtracting  once  the  multiplicand  from  the 

327  result  we  get  the  desired  product. 
12753   Ans. 

9.    To  multiply  by  15,  150,  and  1500. 
Multiply  324  by  15. 

Explanation. — Annex  a  cipher  to  the  multiplicand, 

OPKHATION" 

take  one  half  of  that  number  and  add  to  it  and  you  have 

the  desired  product. 
1^=^  To  multiply  by  150,  annex  two  ciphers,  and  to  multiply 

4860  Ans.  ^^  ^^^^  axvne^  three  ciphers. 

10.    To  multiply  two  numbers  ending  in  5. 

Rule.  —  To  multiply  two  small  numbers  each  ending  in  5,  such 


SHORT   ^METHODS  233 

as  S5  and  75,  take  the  product  of  the  left-hand  figures  (the  S  and 
7),  increased  by  Imlf  their  sunij  and  prefix  the  result  to  25. 

Thus  35  5  X  5  =  25. 

16  3  X  7  4- 1(3  +  ")  =  26. 

2625,  Ans. 

11.  To  square  any  number  of  two  digits. 

Rule.  —  Square  the  figure  in  units^  place  to  obtain  the  figure 
ill  units^  place  of  the  answer  and  carry  as  in  multiplication. 
TJien  take  twice  tJie  product  of  the  figures  in  units'  and  tens* 
])lace,  plus  the  amount  carried.  To  the  jjart  of  the  square 
thus  far  obtained  prefix  the  square  of  the  figure  in  tens'  place 
plus  the  amount  carried. 

Thus  (84)2  =  7056. 

4^  =  16.     Put  down  6  and  carry  1. 
2  (8  X  4)  -h  1  =  65.     Put  down  5  and  carry  6. 
82  +  6  =  70.     Prefix  70  to  56. 
This  also  applies  to  numbers  of  more  than  two  digits,  though 
not  so  readily  performed  mentally, 

12.  To  square  a  number  ending  in  5. 

Rule.  —  To  square  a  number  ending  in  5,  such  as  85,  take 
the  product  of  8  by  the  next  higher  figure  (9)  and  annex  25  to 
the  result. 

Thus  85-  =  7225. 

13.  To  square  any  number  consisting  of  9's. 

Rule.  —  Write  as  many  9*s  less  one  as  there  are  in  the  given 
number,  an  8,  as  unany  ciphers  as  9*s,  and  a  I. 

Thus  9992  =  998001. 

14.  To  multiply  by  complements.  Complements  are  useful 
not  only  in  addition  and  subtraction,  but  also  in  multiplication. 
When  the  complements  are  small  and  the  numbers  of  which 
they  are  complements  are  large,  there  is  a  great  advantage  in 
this  method. 


234  MATHEMATICAL   WRINKLES 

Multiply  98  by  95. 

OPERATION  Explanation. — The  product  of  the  comple- 

98   complement  2  merits  gives  the  two  right-hand  figures,  10,  and 

95  complement  _5_  subtracting  either  complement  from  the  other  fac- 

9310                         10  tor  gives  the  other  two  figures,  93. 

Multiply  198  by  192. 

Explanation.  —  When  the   numbers    to    be 

OPERATION  ,..    1.     1  ,     ,  ,         1       , 

multiplied   are   between   one  hundred  and  two 

,^-  ,  ^   -        hundred,   the  remainder  found  by  subtracting 

192  complement  8         .,,  '  ,  .        .u      .u  v.  Z. 

..,-3-, ,  —      either  complement  from  the  other  number  must 

be  doubled. 

Note.  —  If  the  numbers  to  be  multiplied  are  between  two  hundred  and 
three  hundred,  the  remainder  must  be  multiplied  by  three  ;  between  three 
hundred  and  four  hundred  by  four ;  between  four  hundred  and  five  hun- 
dred by  five  ;  and  so  on. 

15.   To  multiply  by  excesses. 

Rule.  —  From  the  sum  of  the  numbers  subtract  100  or  1000,  as 
required,  and  annex  the  product  of  the  excesses. 

Note.  — An  excess  is  the  amount  greater  than  100,  1000,  etc. 

Example.  —  112  x  103  =  11536. 

112  +  03  =  115. 
To  115  annex  12  x  3,  or  36  =  11536. 
Example.  —  1009  x  1007  =  1016063. 

1009  +  007  =  1016. 
To  1016  annex       063  =  1016063. 

DIVISION 

When  the  divisor  is  an  aliquot  part  of  some  higher  unit. 

1.  To  divide  by  2\,  multiply  the  dividend  by  4  and  point  off 
one  place. 

2.  To  divide  by  5,  multiply  the  dividend  by  2  and  point  off 
one  place. 

3.  To  divide  by  10,  point  off  one  place. 


SHORT   METHODS  235 

4.  To  divide  by  12^,  multiply  the  dividend  by  8  and  point 
off  two  places. 

5.  To  divide  by  16|,  multiply  the  dividend  by  6  and  point 
off  two  places. 

6.  To  divide  by  20,  multiply  the  dividend  by  5  and  point 
off  two  places. 

7.  To  divide  by  25,  multiply  the  dividend  by  4  and  point 
off  two  places. 

8.  To  divide  by  33J,  multiply  the  dividend  by  3  and  point 
off  two  places. 

9.  To  divide  by  50,  multiply  the  dividend  by  2  and  point 
off  two  places. 

10.  To  divide  by  66|,  multiply  the  dividend  by  3,  point  off 
two  places,  and  divide  by  2. 

11.  To  divide  by  100,  point  off  two  places. 

12.  To  divide  by  125,  multiply  the  dividend  by  8  and  point 
off  three  places. 

13.  To  divide  by  200,  multiply  the  dividend  by  5  and  point 
off  three  places. 

14.  To  divide  by  250,  multiply  the  dividend  by  4  and  point 
off  three  places. 

15.  To  divide  by  500,  multiply  the  dividend  by  2  and  point 
off  three  places. 

16.  To  divide  by  1000,  point  off  three  places. 

FRACTIONS 

1.    To  add  two  fractions  which  have  1  for  their  numerator. 
Rule.  —  Write  the  sum  of  the  given  denominators  over  the  prod- 
uct of  the  given  denominators. 

Thus  i  +  i  =  jV 


236  MATHEMATICAL   WKINKLES 

2.  To  subtract  two  fractions  which  have  1  for  their  numerator. 
Rule. —  Write  the  difference  of  the  given  denominators  over  the 

product  of  the  given  denominators. 

±nus  ^       -g-  _  2^-j.  _  ^^. 

3.  To  multiply  two  mixed  numbers  when  the  whole  numbers 
are  the  same  and  the  sum  of  the  fractions  is  1. 

Rule.  —  Multiply  the  lohole  number  by  the  next  highest  whole 
number  J  and  to  the  product  thus  obtained  add  the  product  of  the 
fractions. 

Thus  94  X  91  =  9O2V 

4.  To  multiply  two  mixed  numbers  when  the  difference  of 
the  whole  numbers  is  1,  and  the  sum  of  the  fractions  is  1. 

Rule.  —  Multiply  the  larger  number  increased  by  1,  by  the 
smaller  number;  then  square  the  fraction  belonging  to  the  larger 
mimber  and  subtract  its  square  from  1.  Add  the  whole  number 
and  the  fraction  and  you  have  the  desired  product. 

Thus  54  X  44  =  24Jt. 

5.  To  multiply  two  mixed  numbers  ending  in  J. 

Rule.  —  To  the  product  of  the  whole  numbers,  add  half  their  sum 
plus  \.  (If  the  sum  be  an  odd  number,  call  it  one  less,  to  make  it 
even,  and  annex  |.) 

Thus  81  X  64  =  ^b\,  ^x^  =  35f ,  etc. 

6.  To  square  any  number  ending  in  one  half. 

Rule.  —  Midtiply  the  number  by  itself  increased  by  unity,  and 
annex  \. 

7.  To  square  any  number  ending  in  one  fourth. 

Rule.  —  Multiply  the  number  by  itself  increased  by  ^,  and  annex 

8.  To  square  any  number  ending  in  three  fourths. 

Rule.  —  Multiply  the  number  by  itself  increased  by  1-|-,  and 
annex  ^^. 


SHORT  METHODS  237 

9.    To  square  any  number  ending  in  one  third. 
Rule.  —  Multiply  the   number  by  itself  increased   by  J,   and 
annex  J. 

10.  To  square  any  number  ending  in  two  thirds. 

Rule.  —  Multiply  the  number  i>y  itself  increased  by  1\,  and 
annex  ^. 

11.  To  multiply  two  numbers  ending  with  the  same  fraction. 
Rule.  —  To  the  product  of  the  whole  numbers^  add  that  fraction 

of  their  sum,  and  the  square  of  the  fraction. 

Thus  lof  X  6f  =  90  -f  6  H-  A  =  96:^. 

12.  To  square  any  mixed  number. 

Rule.  —  Multiply  the  whole  number  by  itself  increased  by  twice 
the  fraction,  and  add  the  square  of  the  frojctimx. 

INTEREST 

1.  The  Thirty-six  Per  Cent  Method. 

Rule.  —  Multiply  the  principal  by  the  time  in  days,  move  the 
decimal  point  three  plox^es  to  the  left,  and  divide: 

If  at  1  %  by  36.  If  at    7  %  by  5.143. 

If  at  2  %  by  18.  If  at    8  %  by  4.5. 

If  at  3  %  by  12.  If  at    9  %  by  4. 

If  at  4  %  by  9.  If  at  10  %  by  3.6. 

If  at  5  %  by  7.2.  If  at  11  %  by  3.273. 

If  at  6%  by  6.  If  at  12%  by  3. 

2.  The  Bankers'  Sixty-day  Method. 

Rule.  —  (a)  Moving  the  decimal  point  in  the  principal  three 
places  to  the  left  gives  the  interest  ai  6  fo  for  6  days. 

Moving  the  decimal  point  in  the  principal  two  places  to  the  left 
gives  the  interest  at  6%  for  60  days. 

Moving  the  decimal  point  in  the  principal  one  place  to  the  left 
gives  the  interest  at  6%  for  600  days. 


238  MATHEMATICAL   WRINKLES 

Writing  the  principal  for  the  interest  gives  the  interest  at  6  (Jo 
for  6000  days. 

(h)  The  interest  for  any  other  time  or  rate  can  easily  be  found 
by  using  convenient  multiples  or  aliquot  parts. 

Thus  Interest  on  $36  for        6  days  at  6  %  =  $     .036. 
Interest  on  $  36  for      60  days  at  6  %  =  S     .36. 
Interest  on  $36  for    600  days  at  6  %  =  $    3.60. 
Interest  on  $  36  for  6000  days  at  6  %  =  $  36.00. 

Example.  —  Find  the  interest  on  $  300  for  4  yr.  6  mo.  18  da. 
at  6%. 

OPERATION 

$72.00  =  interest  for  the  number  of  years. 

$  9.00  =  interest  for  the  number  of  months, 

$     .90  =  interest  for  the  number  of  days. 

$81.90  =  the  required  interest. 
Explanation.  —  6  %  of  $300  =  $  18,  the  interest  for  one  year.     4  x  $  18 
=  $72,  the  interest  for  4  years.     $3  =  the  interest  for  2  months.     3  x  $3 
=  $9,  the  interest  for  6  months.     3  x  $  .30  =  $  .90,  the  interest  for  18  days. 

3.    The  Six  Per  Cent  Method. 

Interest  on  $  1  for  1  year     =  $  .06. 
Interest  on  $1  for  1  month  =  $  .OOJ. 
Interest  on  $  1  for  1  day      =  $  .OOOi. 

Rule.  — Multiply  6  cents  by  the  7iuniber  of  years,  \  a  cent  by  the 
number  of  months,  ^  of  a  mill  by  the  number  of  days,  and  multi- 
ply their  sum  by  the  principal. 

Example.  —  Find  the  interest  on  $400  at  6  %  for  6  yr. 
4  mo.  12  da. 

OPERATION 

$  .36    =  interest  on.$  1  for  number  of  years. 

.02    =  interest  on  $  1  for  number  of  months. 

.002  =  interest  on  $  1  for  number  of  days. 
$.382  =  interest  on  $1  for  the  given  time. 
400 


$152.80  =  the  required  interest. 


SHORT   METHODS 


239 


4.   The  Cancellation  Method. 

(1)  When  the  time  is  in  years. 

Formula : 

J  ■  _  Principal  X  Rate  X  Time 

(2)  When  the  time  is  in  months. 

Formula : 

T  ^       .      Principal  x  Rate  x  Time 
^"*^'^^'  = 100102 

(3)  When  the  time  is  in  days. 

Formula : 

T  t       t  —  ^^^"c^P^^  X  ^^^^  X  Time 
n  eres  -  ^^^  ^  ^^^ 

Exact  Interest  =  Principal  x  Rate  x  Time, 
100  X  365 


OPERATION 


Example. —  Find  the  interest 
on  SOOO  at  12%  for  1  year, 
3  months,  12  days. 


^L2iiL>L462^  ^02.40,  interest. 

5 

5.   The  New  Cancellation  Method. 

Rule.  —  Wnte  the  principal,  timey  and  rate  at  the  right  of  a 
vertical  line;  at  the  lejl  of  this  line  write  a  year  in  the  same  de- 
nomination in  which  the  time  is  expressed.  Cancel  and  reduce. 
Tlie  result  will  be  the  interest  for  the  given  time  and  rate. 

OPERATION 


Example. — Find  the  interest  on  $  1080  for 
3  yr.  4  mo.  12  da.  at  6  %. 


Example. —  Find  the  interest  on  $  540  for 
2  yr.  4  mo.  12  da.  at  10  %. 


$    90 

im 

;? 

40.4 

.06 

$218.16  =  interest. 

OPERATION 
6 


213 

m 

.10 


$127.80  =  interest. 


240  MATHEMATICAL    WRINKLES 

6.  The  Cancellation-Thirty-six  Per  Cent  Method. 
Formula : 

Interest  =  -QQl  Qf  Principal  x  Number  of  Days  x  Rate 

36 

This  method  is  a  combination  of  the  Cancellation  Method 
and  Thirty-six  Per  Cent  Method  and  should  be  very  popular 
on  account  of  its  simplicity. 

OPERATION 

Example.  —  Find  the  interest      '^* 
on  S5112  at  4  %  for  100  days.  3^  =$56.80,  interest. 

9 

7.  The  Twelve  Per  Cent  Method. 

To  find  the  interest  for  1  month  on  any  principal  at  12  %, 
simply  remove  the  decimal  point  two  places  to  the  left  in  the 
principal ;  in  other  words,  divide  the  principal  by  100.  This 
gives  the  interest  for  1  month  at  12  %. 

Rule.  —  Poi7}t  off  two  places  in  the  principal,  and  multiply  by  the 
time  expressed  in  months  and  decimals,  or  fractions  of  a  month. 

Example.— What   is  the  operation 

interest   on    $185    at    12%        $1.85  =  interest  at  12%  for  1  month. 

for  3  months,  15  days  ?  ^^^^'"^^  '"^  T""!^"' 

-^  $6.47|  =  interest  for  3^  months,  Ans. 


APPROXIMATE  RESULTS 

In  scientific  investigations  exact  results  are  rarely  possible, 
since  the  numbers  used  are  obtained  by  observation  or  by 
experiments  and  are  only  approximate.  There  is  a  degree  of 
accuracy  beyond  which  it  is  impossible  to  go. 

The  student  should  always  bear  in  mind  that  it  is  a  waste 
of  time  to  carry  out  results  to  a  greater  degree  of  accu- 
racy than  the  data  on  which  they  are  founded.  Results 
beyond  two  or  three  decimal  places  are  seldom  desired  in 
business. 


SHORT   METHODS  241 

I.  Multiplication. 

Rule.  —  I.  Write  the  terms  of  the  multiplier  in  a  reverse  order, 
placing  the  units'  term  under  that  tei-vi  of  the  multiplicand  which 
is  of  the  lowest  order  m  the  required  product. 

II.  Multiply  each  term  of  the  multiplicand  by  the  multiplier y 
rejecting  those  terms  that  are  on  the  right  of  the  term  used  as  a 
multiplier,  increasing  each  partial  product  by  as  many  units  as 
would  have  been  caiTied  to  it  from  the  product  of  the  rejected  part 
of  the  multiplicand,  and  one  more  when  the  second  term  toioards 
the  light  in  the  product  of  the  rejected  terms  is  5  or  more  than  5 ; 
and  place  the  right-hand  terms  of  these  partial  products  in  the 
same  column. 

III.  Add  the  partial  products,  and  point  off  in  the  sum  the 
required  number  of  decimal  places. 

OPERATION 

4.78567 
95141.3 

Example.— Multiply  4.78567         14.3570  =  4.7856  x  3  +  .0002. 

by   3.14159,   correct   to   four  'I'^  = '''' ^  !  + -^i* 

/.       ,      ,'  .1914  =  4.78  X  .04 +  .0002. 

decimal  places.  48  =  4.7  x  .001  +  .0001. 

24  =  4  X  .0005  +  .0004. 
4  =  0+  .00009  +  .0004. 


15.0346 


2.   Division. 

Rule.  —  I.  Compare  the  divisor  with  the  dividend  to  ascertain 
the  number  of  terms  in  the  quotient. 

II.  For  the  first  contracted  divisor,  take  as  many  terms  of  the 
divisor,  beginning  with  the  first  significant  term  on  the  left,  as 
there  are  terms  in  the  quotient;  and  for  each  successive  divisor, 
reject  the  right-hand  term  of  the  previous  divisor,  until  all  the 
terms  of  the  divisor  have  been  rejected. 

III.  In  multiplying  by  the  several  terms  of  the  quotient,  carry 
from  the  rejected  terms  of  tJie  divisor  as  in  contracted  multiplica- 
tion. 


242 


MATHEMATICAL   WEINKLES 


Example.  —  Divide  35.765342 
by  8.76347,  correct  to  four  deci- 
mal places. 


OPERATION 

8.76347)35.765342(4.0811 

35  0539  =  4  X  87634  + 
7114 

7010:^8  X  876  +  2 
104 

88  =  1  X  87  +  1 
16 

9=1x8+1 
7 


3.    Square  Root. 

Rule.  —  Find,  as  visual,  more  than  one-half  the  terms  of  the 
root,  and  then  divide  the  last  remainder  by  the  last  divisor,  using 
the  contracted  method. 

Example.  —  Extract  the  square  root  of  10. 


61 


OPERATION 

10(3.16227766+ 
9 

100 
61 


626 


3900 
3756 


6322 


14400 
12644 


63242 


175600 
126484 


632447 


4911600 
4427129 


6324547 


48447100 
44271820 


63245546 


417527100 
379473276 


632455526 


3805382400 
3794733156 


CONTRACTED    METHOD 

10(3.16227766+ 
9 


61  1 

100 

61 

626 

3900 

3756 

6322 

14400 

12644 

63242 


175600 

126484 

49116 

44269 

4847 

4427 

"420 

379 

"41 

38 


4.    Cube  Eoot. 

Rule.  —  Extract  the  cube  root,  as  usual,  until  one  more  than 
half  the  terms  required  in  the  root  have  been  found;  then  with 


SHORT  METHODS  243 

the  trial  divisor  and  last  remainder  proceed,  as  in  contracted 
division  of  decimals,  to  find  the  other  terms  of  the  root,  dropping 
two  figures  instead  of  one  from  the  divisor  at  each  step,  and  one 
from  each  remainder. 

Example.  —  Extract   tlie   cube   root   of  2  to   four   decimal 
places. 


OPERATIOX 

2.000000  1  1.2599 
1 

300 

60 

4 

1000 

304 

728 

43200 

1800 

25 

272000 

45025 

225125 

Next  trial  divisor,  40i5T^  |  4687^  remainder. 

4219  =  9  X  408  +  9  X  75 
46^ 
_42  =  4x  9  +  6 
4 

6.   Extraction  of  Any  Root. 

Rule.  —  Obtain  one  less  than  half  of  the  figures  required  in  the 
root  as  the  nde  directs;  then,  instead  of  annexing  ciphers  and 
bringing  down  a  period  to  the  last  numbers  in  the  columns,  leave 
the  remainder  in  the  right-hand  column  for  a  dividend;  cutoff 
the  right-hand  fuf  a  re  from  the  last  number  of  the  j^revious  column, 
two  right-hand  figures  fro7n  the  last  number  in  the  column  before 
that,  and  so  on,  always  cutting  off  one  more  figure  for  every  col- 
umn to  the  left. 

With  the  number  in  the  right-hand  column  and  the  one  in  the 
previous  column,  determine  the  next  figure  of  the  root,  and  use  it 
as  directed  in  the  rule,  recollecting  that  the  figures  cut  off  are  not 
used  except  in  carrying  the  tens  they  produce. 

TJiis  process  is  continued  until  the  required  number  of  figures 


244 


MATHEMATICAL   WRINKLES 


is  obtained,  observing  that  when  all  the  figures  in  the  last  number 
of  any  column  are  cut  off,  that  column  will  be  no  longer  used. 

Remark.  —  Add  to  the  1st  column  mentally  ;  multiply  and  add  to  the 
next  column  in  one  operation  :  multiply  and  subtract  from  the  right-hand 
column  in  like  manner. 

Example.  —  Extract  the  cube  root  of  44.6  to  six  decimals. 


0 

9 

2  700 

3  17  5 
367  500 
37  17  16 
37  594^ 
37  659 
37/^3 


OPERATION 

4  4  .  6  0  0  (3 

17  600 

1725000 

238  136 

12182 

865 

111 


546323 


0 

3 

6 

90 

95 

100 

1050 

1054 

1058 

Remark.  —  The  trial  divisors  may  be  known  by  ending  in  two  ciphers ; 
the  complete  divisors  stand  just  beneath  them.  After  getting  3  figures  of 
the  root,  contract  the  operation  by  last  rule. 

—  From  Ray's  "  Higher  Arithmetic." 

MARKING  GOODS 

To  find  the  selling  price  of  a  single  article  at  a  certain  per 
cent  profit  when  the  price  per  dozen  and  rate  per  cent  gain  are 
given. 

Thus,  to  make  5  per  cent,  multiply  the  cost  per  dozen  by  .08f . 

multiply  by  .11| 
multiply  by  .12-j3- 
multiply  by  .121 
multiply  by  .12|i 
multiply  by  .13^ 
multiply  by  .13} 
multiply  by  .13f 
multiply  by  .14.^ 
multiply  by  .15 
multiply  by  .16-| 


6  %    multiply  by  .OSf 

40% 

8  %    multiply  by  .09 

45% 

10  %    multiply  by  .09^ 

50% 

121  cf^  multiply  by  .09f 

^^% 

15  %    multiply  by  M^^ 

60% 

20  %    multiply  by  .10  " 

65% 

25  %    multiply  by  .lO^^ 

66|^ 

30  %    multiply  by  .lOf 

75% 

33^%  multiply  by  .11^ 

80% 

35  %    multiply  by  .Hi 

100% 

QUOTATIONS   ON  MATHEMATICS 

"  Mathematics,  the  queen  of  the  sciences."  —  Gauss. 

"  Mathematics,  the  science  of  the  ideal,  becomes  the  means 
of  investigating,  understanding,  and  making  known  the  world 
of  the  real."  —  White. 

"  Mathematics  is  the  glory  of  the  human  mind."  —  Leibnitz. 

"  The  two  eyes  of  exact  science  are  mathematics  and  logic."  — 
De  Morgan. 

"  Mathematics  is  the  science  which  draws  necessary  conclu- 
sions from  given  premises."  —  Pierce. 

"  The  advance  and  the  perfecting  of  mathematics  are  closely 
joined  to  the  prosperity  of  the  nation."  —  Napoleon. 

"  Geometry  is  the  perfection  of  logic,  and  excels  in  training 
the  mind  to  logical  habits  of  thinking.  In  this  respect  it  is 
superior  to  the  study  of  logic  itself,  for  it  is  logic  embodied  in 
the  science  of  tangible  form."  —  Brooks. 

"God  geoiuetrizes  continually,"  was  Plato's  reply  when 
questioned  as  to  the  occupation  of  the  Deity. 

"  There  is  no  royal  road  to  geometry."  —  Euclid. 

"  Let  no  one  who  is  unacquainted  with  geometry  enter  here," 
was  the  inscription  over  the  entrance  into  the  academy  of  Plato 
the  philosopher. 

"  All  scientific  education  which  does  not  commence  with 
mathematics  is,  of  necessity,  defective  at  its  foundation."  — 

COMTE. 

"  A  natural  science  is  a  science  only  in  so  far  as  it  is  mathe- 
matical." —  Kant. 

246 


246  MATHEMATICAL   WRINKLES 

"Mathematics  is  the  language  of  definiteness,  the  necessary 
vocabulary  of  those  who  know."  —  White. 

"The  laws  of  nature  are  but  the  mathematical  thoughts  of 
God."  —  Kepler. 

"  Mathematics  is  the  most  marvelous  instrument  created  by 
the  genius  of  man  for  the  discovery  of  truth."  — Laisant. 

"  Euclid  has  done  more  to  develop  the  logical  faculty  of  the 
world  than  any  book  ever  written.  It  has  been  the  inspiring 
influence  of  scientific  thought  for  ages,  and  is  one  of  the 
cornerstones  of  modern  civilization."  —  Brooks. 

"Mathematics  is  thinking  God's  thought  after  Him. 
When  anything  is  understood,  it  is  found  to  be  susceptible  of 
mathematical  statement.  The  vocabulary  of  mathematics  is 
the  ultimate  vocabulary  of  the  material  universe."  —  White. 

"  Geometry  is  regarded  as  the  most  perfect  model  of  a  de- 
ductive science,  and  is  the  type  and  model  of  all  science."  — 
Brooks'  "Mental  Science." 

"  I  have  always  treated  and  considered  puzzles  from  an  edu- 
cational standpoint,  for  the  reason  that  they  constitute  a  species 
of  mental  gymnastics  which  sharpen  the  wits,  clear  fog  and 
cobwebs  from  the  brain,  and  school  the  mind  to  concentrate 
properly.  Comparatively  but  few  people  know  how  to  think 
properly.  As  a  school  for  mechanical  ingenuity,  for  stirring 
up  the  gray  matter  in  the  brain,  puzzle  practice  stands  unique 
and  alone."  —  Sam  Loyd. 

"  Geometry  not  only  gives  mental  power,  but  it  is  a  test  of 
mental  power.  The  boy  who  cannot  readily  master  his 
geometry  will  never  attain  to  much  in  the  domain  of  thought. 
He  may  have  a  fine  poetic  sense  that  will  make  a  writer  or  an 
orator;  but  he  can  never  reach  any  eminence  in  scientific 
thought  or  philosophic  opinion.  AH  the  great  geniuses  in  the 
realm  of  science,   as   far   as   known,   had  fine  mathematical 


QUOTATIONS   ON   MATHEMATICS  247 

abilities.  So  valuable  is  geometry  as  a  discipline  that  many 
lawyers  and  preachers  review  their  geometry  every  year  in 
order  to  keep  the  mind  drilled  to  logical  habits  of  thinking." 
—  Brooks*  "  Mental  Science." 

"Mathematics  is  the  very  embodiment  of  truth.  No  true 
devotee  of  mathematics  can  be  dishonest,  untruthful,  unjust. 
Because,  working  ever  with  that  which  is  true,  how  can  one 
develop  in  himself  that  which  is  exactly  opposite  ?  It  would 
be  as  though  one  who  was  always  doing  acts  of  kindness  should 
develop  a  mean  and  groveling  disposition.  Mathematics,  there- 
fore, has  ethical  value  as  well  as  educational  value.  Its  prac- 
tical value  is  seen  about  us  everyday.  To  do  away  with  every 
one  of  the  many  conveniences  of  this  present  civilization  in 
which  some  mathematical  principle  is  applied,  would  be  to 
turn  the  finger  of  time  back  over  the  dial  of  the  ages  to  the 
time  when  man  dwelt  in  caves  and  crouched  over  the  bodies 
of  wild  beasts."  —  B.  F.  Fixkel. 

"  As  the  drill  will  not  penetrate  the  granite  unless  kept  to 
the  work  hour  after  hour,  so  the  mind  will  not  penetrate  the 
secrets  of  mathematics  unless  held  long  and  vigorously  to  the 
work.  As  the  sun's  rays  burn  only  when  concentrated,  so  the 
mind  achieves  mastery  in  mathematics,  and  indeed  in  every 
branch  of  knowledge,  only  when  its  possessor  hurls  all  his 
forces  upon  it.  Mathematics,  like  all  the  other  sciences,  opens 
its  door  to  those  only  who  knock  long  and  hard.  No  more 
damaging  evidence  can  be  adduced  to  prove  the  weakness  of 
character  than  for  one  to  have  aversion  to  mathematics ;  for 
whether  one  wishes  so  or  not,  it  is  nevertheless  true,  that  to 
have  aversion  for  mathematics  means  to  have  aversion  to  ac- 
curate, painstaking,  and  persistent  hard  study,  and  to  have 
aversion  to  hard  study  is  to  fail  to  secure  a  liberal  education, 
and  thus  fail  to  compete  in  that  fierce  and  vigorous  struggle 
for  the  highest  and  the  truest  and  the  best  in  life  which  only 
the  strong  can  hope  to  secure."  —  B.  F.  Finkel. 


248  MATHEMATICAL   WRINKLES 

"Mathematics  develops  step  by  step,  but  its  progress  is 
steady  and  certain  amid  the  continual  fluctuations  and  mis- 
takes of  the  human  mind.  Clearness  is  its  attribute,  it  combines 
disconnected  facts  and  discovers  the  secret  bond  that  unites 
them.  When  air  and  light  and  the  vibratory  phenomena  of 
electricity  and  magnetism  seem  to  elude  us,  when  bodies  are 
removed  from  us  into  the  infinitude  of  space,  when  man  wishes 
to  behold  the  drama  of  the  heavens  that  has  been-  enacted  cen- 
turies ago,  when  he  wants  to  investigate  the  effects  of  gravity 
and  heat  in  the  deep,  impenetrable  interior  of  our  earth,  then 
he  calls  to  his  aid  the  help  of  mathematical  analysis.  Mathe- 
matics renders  palpable  the  most  intangible  things,  it  binds  the 
most  fleeting  phenomena,  it  calls  down  the  bodies  from  the  in- 
finitude of  the  heavens  and  opens  up  to  us  the  interior  of  the 
earth.  It  seems  a  power  of  the  human  mind  conferred  upon 
us  for  the  purpose  of  recompensing  us  for  the  imperfection  of 
our  senses  and  the  shortness  of  our  lives.  Nay,  what  is  still 
more  wonderful,  in  the  study  of  the  most  diverse  phenomena 
it  pursues  one  and  the  same  method,  it  explains  them  all  in 
the  same  language,  as  if  it  were  to  bear  witness  to  the  unity 
and  simplicity  of  the  plan  of  the  universe.'^  —  Fourier. 

"  The  practical  applications  of  mathematics  have  in  all  ages 
redounded  to  the  highest  happiness  of  the  human  race.  It 
rears  magnificent  temples  and  edifices,  it  bridges  our  streams 
and  rivers,  it  sends  the  railroad  car  with  the  speed  of  the  wind 
across  the  continent;  it  builds  beautiful  ships  that  sail  on 
every  sea ;  it  has  constructed  telegraph  and  telephone*  lines  and 
made  a  messenger  of  something  known  to  mathematics  alone 
that  bears  messages  of  love  and  peace  around  the  globe ;  and 
by  these  marvelous  achievements,  it  has  bound  all  the  nations 
of  the  earth  in  one  common  brotherhood  of  man." 

B.   F.   FiNKEL. 

"  Mathematics  is  the  indispensible  instrument  of  all  physical 
research." — Berthelot. 


QUOTATIONS  ON  MATHEMATICS  249 

"  It  is  in  mathematics  we  ought  to  learn  the  general  method 
always  followed  by  the  human  mind  in  its  positive  researches." 

—  COMTE. 

"  All  my  physics  is  nothing  else  than  geometry." 

—  Descartes. 

"If  the  Greeks  had  not  cultivated  conic  sections,  Kepler 
could  not  have  superseded  Ptolemy."  —  Whewell. 

"  There  is  nothing  so  prolific  in  utilities  as  abstractions." 

—  Faraday. 
"  I  am  sure  that  no  subject  loses  more  than  mathematics  by 
any  attempt  to  dissociate  it  from  its  history."  —  Glaisher. 

"The  history  of  mathematics  is  one  of  the  large  windows 
through  which  the  philosophic  eye  looks  into  past  ages  and 
traces  the  line  of  intellectual  development." — Cajori. 

"  If  we  compare  a  mathematical  problem  with  a  huge  rock, 
into  the  interior  of  which  we  desire  to  penetrate,  then  the 
work  of  the  Greek  mathematicians  appears  to  us  like  that  of 
a  vigorous  stonecutter  who,  with  chisel  and  hammer,  begins 
with  indefatigable  perseverance,  from  without,  to  crumble  the 
rock  slowly  into  fragments;  the  modern  mathematician  ap- 
pears like  an  excellent  miner  who  first  bores  through  the  rock 
some  few  passages,  from  which  he  then  bursts  it  into  pieces 
with  one  powerful  blast,  and  brings  to  light  the  treasures 
within."  — Hankel. 

"The  world  of  ideas  which  mathematics  discloses  or  illu- 
minates, the  contemplation  of  divine  beauty  and  order  which 
it  induces,  the  harmonious  connection  of  its  parts,  the  infinite 
hierarchy  and  absolute  evidence  of  truths  with  which  mathe- 
matical science  is  concerned,  these,  and  such  like,  are  the 
surest  grounds  of  its  title  to  human  regard."  —  Sylvester. 

"  I  often  find  the  conviction  forced  upon  me  that  the  increase 
of  mathematical  knowledge  is  a  necessary  condition  for  the 
advancement  of  science,  and  if  so,  a  no  less  necessary  condition 


250  MATHEMATICAL   WRINKLES 

for  the  improvement  of  mankind,  I  could  not  augur  well  for 
the  enduring  intellectual  strength  of  any  nation  of  men,  whose 
education  was  not  based  on  a  solid  foundation  of  mathematical 
learning  and  whose  scientific  conception,  or  in  other  words, 
whose  notions  of  the  world  and  of  things  in  it,  were  not  braced 
and  girt  together  with  a  strong  framework  of  mathematical 
reasoning."  —  H.  J.  Stephen  Smith. 

"  If  the  eternal  and  inviolable  correctness  of  its  truths  lends 
to  mathematical  research,  and  therefore  also  to  mathematical 
knowledge,  a  conservative  character  on  the  other  hand,  by  the 
continuous  outgrowth  of  new  truths  and  methods  from  the 
old,  progressiveness  is  also  one  of  its  characteristics.  In  mar- 
velous profusion  old  knowledge  is  augmented  by  new,  which 
has  the  old  as  its  necessary  condition,  and,  therefore,  could  not 
have  arisen  had  not  the  old  preceded  it.  The  indestructibility 
of  the  edifice  of  mathematics  renders  it  possible  that  the  work 
can  be  carried  to  ever  loftier  and  loftier  heights  without  fear 
that  the  highest  stories  shall  be  less  solid  and  safe  than  the 
foundations,  which  are  the  axioms,  or  the  lower  stories,  which 
are  the  elementary  propositions.  But  it  is  necessary  for  this 
that  all  the  stones  should  be  properly  fitted  together ^  and  it 
would  be  idle  labor  to  attempt  to  lay  a  stone  that  belonged 
above  in  a  place  below."  —  Schubert. 

"As  the  sun  eclipses  the  stars  by  his  brilliancy,  so  the  man 
of  knowledge  will  eclipse  the  fame  of  others  in  assemblies  of 
the  people  if  he  proposes  algebraic  problems,  and  still  more  if 
he  solves  them."  —  Brahmagupta. 

"  Mathematical  reasoning  may  be  employed  in  the  inductive 
sciences;  indeed  some  of  their  greatest  achievements  have  been 
obtained  through  mathematics.  By  it  Newton  demonstrated 
the  truth  of  the  theory  of  gravitation;  by  it  Leverrier  dis- 
covered a  new  planet  in  the  heavens ;  by  it  the  exact  time  of 
an  eclipse  of  the  sun  or  moon  is  predicted  centuries  before  it 
comes  to  pass.     Mathematics  is  the  instrument  by  which  the 


QUOTATIONS   ON   MATHEMATICS  251 

engineer  tunnels  our  mountains,  bridges  our  rivers,  constructs 
our  aqueducts,  erects  our  factories  and  makes  them  musical 
with  the  busy  hum  of  spindles.  Take  away  the  results  of  the 
reasoning  of  mathematics,  and  there  would  go  with  it  nearly 
all  the  material  achievements  which  give  convenience  and 
glory  to  modern  civilization."  —  Brooks'  "  Mental  Science  and 
Culture." 

"  The  science  of  geometry  came  from  the  Greek  mind  almost 
as  perfect  as  Minerva  from  the  head  of  Jove.  Beginning  with 
definite  ideas  and  self-evident  truths,  it  traces  its  way,  by  the 
processes  of  deduction,  to  the  profoundest  theorem.  For  clear- 
ness of  thought,  closeness  of  reasoning,  and  exactness  of  truths, 
it  is  a  model  of  excellence  and  beauty.  It  stands  as  a  type  of 
all  that  is  best  in  the  classical  culture  of  the  thoughtful  mind 
of  Greece.  Geometry  is  the  perfection  of  logic ;  Euclid  is  as 
classic  as  Homer."  —  Brooks'  "  Philosophy  of  Arithmetic." 

"Only  a  limited  number  of  people  are  capable  of  appreciat- 
ing the  beauties  of  this  oldest  of  all  sciences."  —  Locke. 

"  The  value  of  mathematical  instruction  as  a  preparation  for 
those  more  difficult  investigations  consists  in  the  applicability, 
not  of  its  doctrines,  but  of  its  methods.  Mathematics  will 
ever  remain  the  past-perfect  type  of  the  deductive  method  in 
general ;  and  the  applications  of  mathematics  to  the  simpler 
branches  of  physics  furnish  the  only  school  in  which  philoso- 
phers can  effectually  learn  the  most  difficult  and  important 
portion  of  their  art,  the  employment  of  the  laws  of  simpler 
phenomena  for  explaining  and  predicting  those  of  the  more 
complex.  These  grounds  are  quite  sufficient  for  deeming  mathe- 
matical training  an  indispensable  basis  of  real  scientific  educa- 
tion, and  regarding,  with  Plato,  one  who  is  dyew/u-cTpryTos,  as 
wanting  in  one  of  the  most  essential  qualifications  for  the  suc- 
cessful cultivation  of  the  higher  branches  of  philosophy."  — 
From  J.  S.  Mill's  "  Systems  of  Logic." 


252  MATHEMATICAL   WRINKLES 

"  Hold  nothing  as  certain  save  what  can  be  demonstrated." 
—  Newton. 

"  To  measure  is  to  know."  —  Kepler. 

"  It  may  seem  strange  that  geometry  is  unable  to  define  the 
terms  which  it  uses  most  frequently,  since  it  defines  neither 
movement,  nor  number,  nor  space  —  the  three  things  with  which 
it  is  chiefly  concerned.  But  we  shall  not  be  surprised  if  we 
stop  to  consider  that  this  admirable  science  concerns  only  the 
most  simple  things,  and  the  very  quality  that  renders  these 
things  worthy  of  study  renders  them  incapable  of  being  de- 
fined. Thus  the  very  lack  of  definition  is  rather  an  evidence 
of  perfection  than  a  defect,  since  it  comes  not  from  the  ob- 
scurity of  the  terms,  but  from  the  fact  that  they  are  so  very 
well  known."  —  Pascal. 

"  The  method  of  making  no  mistake  is  sought  by  every  one. 
The  logicians  profess  to  show  the  way,  but  the  geometers  alone 
ever  reach  it,  and  aside  from  their  science  there  is  no  genuine 
demonstration."  —  Pascal. 

"  We  may  look  upon  geometry  as  a  practical  logic,  for  the 
truths  which  it  studies,  being  the  most  simple  and  most  clearly 
understood  of  all  truths,  are  on  this  account  the  most  suscepti- 
ble of  ready  application  in  reasoning."  — D'Alembekt. 

"  Without  mathematics  no  one  can  fathom  the  depths  of 
philosophy.  Without  philosophy  no  one  can  fathom  the  depths 
of  mathematics.  Without  the  two  no  one  can  fathom  the 
depths  of  anything."  —  Bordas  Demoulin. 

"The  taste  for  exactness,  the  impossibility  of  contenting 
one's  self  with  vague  notions  or  of  leaning  upon  mere  hypoth- 
eses, the  necessity  for  perceiving  clearly  the  connection  between 
certain  propositions  and  the  object  in  view,  —  these  are  the 
most  precious  fruits  of  the  study  of  mathematics."  —  Lacroix. 

"  God  is  a  circle  of  which  the  center  is  everywhere  and  the 
circumference  nowhere."  —  Rabelais. 


QUOTATIONS  ON  MATHEMATICS  253 

"The  sailor  whom  an  exact  observation  of  longitude  saves 
from  shipwreck  owes  his  life  to  a  theory  developed  two  thou- 
sand years  ago  by  men  who  had  in  mind  merely  the  specula- 
tions of  abstract  geometry."  —  Condorcet. 

"  The  statement  that  a  given  individual  has  received  a  sound 
geometrical  training  implies  that  he  has  segregated  from  the 
whole  of  his  sense  impressions  a  certain  set  of  these  impres- 
sions, that  he  has  then  eliminated  from  their  consideration  all 
irrelevant  impressions  (in  other  words,  acquired  a  subjective 
command  of  these  impressions),  that  he  has  developed  on  the 
basis  of  these  impressions  an  ordered  and  continuous  system 
of  logical  deduction,  and  finally  that  he  is  capable  of  express- 
ing the  nature  of  these  impressions  and  his  deductions  there- 
from in  terms  simple  and  free  from  ambiguity.  Now  the 
slightest  consideration  will  convince  any  one  not  already  conver- 
sant with  the  idea,  that  the  same  sequence  of  mental  processes 
underlies  the  whole  career  of  any  individual  in  any  walk  of 
life  if  only  he  is  not  concerned  entirely  with  manual  labor ; 
consequently  a  full  training  in  the  performance  of  such  se- 
quences must  be  regarded  as  forming  an  essential  part  of  any 
education  worthy  of  the  name.  Moreover,  the  full  apprecia- 
tion of  such  processes  has  a  higher  value  than  is  contained  in 
the  mental  training  involved,  great  though  this  be,  for  it  in- 
duces an  appreciation  of  intellectual  unity  and  beauty  which 
plays  for  the  mind  that  part  which  the  appreciation  of  schemes 
of  shape  and  color  plays  for  the  artistic  faculties ;  or,  again, 
that  part  which  the  appreciation  of  a  body  of  religious  doctrine 
plays  for  the  ethical  aspirations.  Now  geometry  is  not  the 
sole  possible  basis  for  inculcating  this  appreciation.  Logic  is 
an  alternative  for  adults,  provided  that  the  individual  is  pos- 
sessed of  sufficient  wide,  though  rough,  experience  on  which  to 
base  his  reasoning.  Geometry  is,  however,  highly  desirable 
in  that  the  objective  bases  are  so  simple  and  precise  that  they 
can  be  grasped  at  an  early  age,  that  the  amount  of  training  for 


254  MATHEMATICAL   WBINKLES 

the  imagination  is  very  large,  that  the  deductive  processes  are 
not  beyond  the  scope  of  ordinary  boys,  and  finally  that  it 
affords  a  better  basis  for  exercise  in  the  art  of  simple  and  exact 
expression  than  any  other  possible  subject  of  a  school  course/' 
—  Carson. 

"  Geometry  is  a  mountain.  Vigor  is  needed  for  its  ascent. 
The  views  all  along  the  paths  are  magnificent.  The  effort  of 
climbing  is  stimulating.  A  guide  who  points  out  the  beauties, 
the  grandeur,  and  the  special  places  of  interest,  commands  the 
admiration  of  his  group  of  pilgrims."  —  David  Eugene  Smith. 

"  If  mathematical  heights  are  hard  to  climb,  the  fundamental 
principles  lie  at  every  threshold,  and  this  fact  allows  them 
to  be  comprehended  by  that  common  sense  which  Descartes 
declared  was  '  apportioned  equally  among  all  men.' "  —  Collet. 

"  The  wonderful  progress  made  in  every  phase  of  life  during 
the  last  hundred  years  has  been  possible  only  through  the 
increasing  use  of  symbols.  To-day,  only  the  common  laborer 
works  entirely  with  the  actual  things.  Those  who  occupy  more 
remunerative  positions  in  the  business  world  work  very  largely 
with  symbols,  and  in  the  professional  world  the  possession  of 
and  ability  to  use  a  set  of  symbols  is  a  prerequisite  of  even 
moderate  success.  The  work  of  a  man's  hands  remains  after 
the  worker  has  gone,  but  the  products  of  mental  labor  are  lost 
unless  they  are  preserved  to  the  world  through  some  symbolic 
medium.  It  may  be  said  without  fear  of  successful  contradic- 
tion that  the  language  of  mathematics  is  the  most  widely  used 
of  any  symbolism.  The  man  who  has  command  of  it  possesses 
a  clear,  concise,  and  universal  language.  Fallacies  in  reason- 
ing and  discrepancies  in  conclusions  are  easily  detected  when 
ideas  are  expressed  in  this  language.  The  most  abstruse  prob- 
lem is  immediately  clarified  when  translated  into  mathematics. 
To  quote  from  M.  Berthelot,  '  Mathematics  excites  to  a  high 
degree  the  conceptions  of  signs  and  symbols  —  necessary  in- 


QUOTATIONS  ON   MATHEMATICS  255 

struments  to  extend  the  power  and  reach  of  the  human  mind 
by  summarizing.  Mathematics  is  the  indispensable  instrument 
of  all  physical  research.'  But  not  only  physical  but  all  scien- 
tific research  must  avail  itself  of  this  same  instrument.  In- 
deed, so  completely  is  nature  mathematical  that  to  him  who 
would  know  nature  there  is  no  recourse  but  to  be  conversant 
with  the  language  of  mathematics."  —  Carpenter. 

No  less  an  astronomer  than  J.  Herschel  has  said  of  as- 
tronomy: "Admission  to  its  sanctuary  and  to  the  privileges 
and  feelings  of  a  votary  is  only  to  be  gained  by  one  means  — 
sound  and  sufficient  knowledge  of  mathematics,  the  great  in- 
strument of  all  exact  inquiry,  without  which  no  man  can  ever 
make  such  advances  in  this  or  any  other  of  the  higher  depart- 
ments of  science  as  can  entitle  him  to  form  an  independent 
opinion  on  any  subject  of  discussion  within  their  range." 

"It  is  only  through  mathematics  that  we  can  thoroughly 
understand  what  true  science  is.  Here  alone  can  we  find  in 
the  highest  degree  simplicity  and  severity  of  scientific  law, 
and  such  abstraction  as  the  human  mind  can  attain.  Any  sci- 
entific education  setting  forth  from  any  other  point  is  faulty 
in  its  basis."  —  Comte. 

"  The  enemies  of  geometry,  those  who  know  it  only  imper- 
fectly, look  upon  the  theoretical  problems,  which  constitute 
the  most  difficult  part  of  the  subject,  as  mental  games  which 
consume  time  and  energy  that  might  better  be  employed  in 
other  ways.  Such  a  belief  is  false,  and  it  would  block  the 
progress  of  science  if  it  were  credible.  But  aside  from  the 
fact  that  the  speculative  problems,  which  at  first  sight  seem 
barren,  can  often  be  applied  to  useful  purposes,  they  always 
stand  as  among  the  best  means  to  develop  and  to  express  all 
the  forces  of  the  human  intelligence." —  Abbe  Bossut. 

"  We  study  music  because  music  gives  us  pleasure,  not  neces- 
sarily our  own  music,  but  good  music,  whether  ours,  or,  as  is 


256  MATHEMATICAL   WEINKLES 

more  probable,  that  of  others.  We  study  literature  because 
we  derive  pleasure  from  books ;  the  better  the  book,  the  more 
subtle  and  lasting  the  pleasure.  We  study  art  because  we  re- 
ceive pleasure  from  the  great  works  of  the  masters,  and  prob- 
ably we  appreciate  them  the  more  because  we  have  dabbled  a 
little  in  pigments  or  in  clay.  We  do  not  expect  to  be  com- 
posers, or  poets,  or  sculptors,  but  we  wish  to  appreciate  music 
and  letters  and  the  fine  arts,  and  to  derive  pleasure  from  them 
and  to  be  uplifted  by  them.  At  any  rate  these  are  the  nobler 
reasons  for  their  study. 

"  So  it  is  with  geometry.  We  study  it  because  we  derive 
pleasure  from  contact  with  a  great  and  an  ancient  body  of 
learning  that  has  occupied  the  attention  of  master  minds  dur- 
ing the  thousands  of  years  in  which  it  has  been  perfected,  and 
we  are  uplifted  by  it.  To  deny  that  our  pupils  derive  this 
pleasure  from  the  study  is  to  confess  ourselves  poor  teachers, 
for  most  pupils  do  have  positive  enjoyment  in  the  pursuit  of 
geometry,  in  spite  of  the  tradition  that  leads  them  to  proclaim 
a  general  dislike  for  all  study.  This  enjoyment  is  partly  that 
of  the  game,  —  the  playing  of  a  game  that  can  always  be  won, 
but  that  cannot  be  won  too  easily.  It  is  partly  that  of  the  aes- 
thetic, the  pleasure  of  symmetry  of  form,  the  delight  of  fitting 
things  together.  But  probably  it  lies  chiefly  in  the  mental  up- 
lift that  geometry  brings,  the  contact  with  absolute  truth,  and 
the  approach  that  one  makes  to  the  Infinite.  We  are  not 
quite  sure  of  any  one  thing  in  biology ;  our  knowledge  of  ge- 
ology is  relatively  very  slight,  and  the  economic  laws  of  society 
are  uncertain  to  every  one  except  some  individual  who  at- 
tempts to  set  them  forth ;  but  before  the  world  was  fashioned 
the  square  on  the  hypotenuse  was  equal  to  the  sum  of  the 
squares  on  the  other  two  sides  of  a  right  triangle,  and  it  will 
be  so  after  this  world  is  dead ;  and  the  inhabitant  of  Mars,  if 
he  exists,  probably  knows  its  truth  as  we  know  it.  The  uplift 
of  this  contact  with  absolute  truth,  with  truth  eternal,  gives 
pleasure  to  humanity  to  a  greater  or  less  degree,   depending 


QUOTATIONS   ON  MATHEMATICS  257 

upon  the  mental  equipment  of  the  particular  individual ;  but 
it  probably  gives  an  appreciable  amount  of  pleasure  to  every 
student  of  geometiy  who  has  a  teacher  worthy  of  the  name." 
—  From  "The  Teaching  of  Geometry,"  by  David  Eugene 
Smith. 

Mathematics  has  not  only  commercial  value,  but  also  educa- 
tional, rhetorical,  and  ethical  value.  No  other  science  offers 
such  a  rich  opportunity  for  original  investigation  and  dis- 
covery. While  it  should  be  studied  because  of  its  practical 
worth,  which  can  be  seen  about  us  every  day,  the  primary 
object  in  its  study  should  be  to  obtain  mental  power,  to 
sharpen  and  strengthen  the  powers  of  thought,  to  give  pen- 
etrating power  to  the  mind  which  enables  it  to  pierce  a  subject 
to  its  core  and  discover  its  elements ;  to  develop  the  power  to 
express  one's  thoughts  in  a  forcible  and  logical  manner;  to 
develop  the  memory  and  the  imagination;  to  cultivate  a  taste 
for  neatness  and  a  love  for  the  good,  the  beautiful,  and  the 
true ;  and  to  become  more  like  the  gi*eatest  of  mathematicians, 
the  Mathematician  of  the  Universe. 


MENSURATION 

Mensuration  is  that  branch  of  mathematics  which  treats  of 
the  measurement  of  geometrical  magnitudes. 

Annulus,  or  Circular  Ring 

An  annul  us  is  the  figure  included  between  two  concentric 
circumferences. 

(1)  To  find  the  area  of  an  annulus. 

Rule.  —  Multiply  the  sum  of  the  two  radii  by  their  difference^ 
and  the  product  by  it. 

Formula.  —  A  =  (i\  +  r^  (r^  —  rg)  tr. 

(2)  To  find  the  area  of  a  sector  of  an  annulus. 

Rule.  —  Multiply  the  sum  of  the  bounding  arcs  by  half  the 
difference  of  their  radii. 

Belts 

Length  of  belts. 

(a)  For  a  crossed  belt, 

L  =  2Vc'-(r,-r,y  +  {r,-r,)f^-2sm-'':i±LA 

(b)  For  an  uncrossed  belt, 

L  =  2^\c^-(r,-r,yi  +  7r(r,  +  r,)  +  2(r,-r,)sm-'''^^, 

where  r^  is  the  greater  radius  and  rj  the  less,  and  c  the  distance 
between  the  parallel  axes. 

258 


MENSURATION  250 

Bins,  Cisterns,  Etc. 

(1)  To  find  the  exact  capacity  of  a  bin  in  bushels. 

Rule.  —  Divide  the  contents  in  cubic  feet  by  .83,  or  {1728-i- 
2150.42);  the  quotient  will  represent  the  number  of  bushels  of 
grain,  etc.  Four  fifths  of  this  number  of  bushels  is  the  number 
of  bushels  of  coal,  apples,  potatoes,  etc.,  that  the  bin  will  hold. 

(2)  To  find  the  approximate  capacity  of  a  bin  in  bushels. 
Rule.  —  Any  number  of  cubic  feet  diminished  by  ^  will  represent 

an  equivalent  number  of  bushels. 

(3)  To  find  the  contents  of  a  cistern,  vessel,  or  space  in 
gallons. 

Rule.  —  Divide  the  contents  in  cubic  inches  by  231  for  liquid 
gallons,  or  by  268.8  for  dry  gallons. 

Brick  and  Stone  Work 

Stonework  is  commonly  estimated  by  the  perch ;  brickwork 
by  the  thousand  bricks. 

(1)  In  estimating  the  work  of  laying  stone,  take  the  entire 
outside  length  in  feet,  thus  measuring  the  corners  twice,  times 
the  height  in  feet,  times  the  thickness  in  feet,  and  divide  by 
24|,  to  obtain  the  number  of  perches.  No  allowance  is  to  be 
made  for  openings  in  the  walls  unless  specified  in  a  written 
contract. 

(2)  In  estimating  the  material  in  stonework,  deduct  for  all 
openings  and  divide  the  exact  number  of  cubic  feet  of  wall  by 
24|,  to  obtain  the  number  of  perches  of  material. 

To  obtain  the  number  of  perches  of  stone,  deduct  ^  for  mor- 
tar and  filling. 

(3)  In  estimating  the  work  of  laying  common  bricks  (com- 
mon bricks  are  8  inches  x  4  inches  x  2  inches  and  22  are  as- 
sumed to  build  1  cubic  foot),  take  the  entire  outside  length  in 
feet,  thus  measuring  the  corners  twice,  times  the  height  in  feet, 
times  the  thickness  in  feet,  and  multiply  by  .022,  to  obtain  the 


260  MATHEMATICAL  WRIKKLES 

number  of  thousand  bricks.     No  allowance  is  to  be  made  for 
openings  in  the  walls  unless  specified  in  a  written  contract. 

(4)  In  estimating  the  material  in  brickwork,  deduct  for  all 
openings  and  multiply  the  exact  number  of  cubic  feet  of  wall 
by  22,  to  obtain  the  number  of  brick  required. 

Carpeting 

Carpets  are  usually  either  1  yard  or  J  yard  in  width. 

The  amount  of  carpet  that  must  be  bought  for  a  room  de- 
pends upon  the  length  and  number  of  strips,  and  the  waste  in 
matching  the  patterns. 

(1)  To  obtain  the  number  of  strips. 

A  fraction  of  a  strip  cannot  be  bought.  Thus,  if  the  num- 
ber of  strips  is  found  to  be  6^,  make  it  7. 

(a)  When  laid  lengthwise. — ^Divide  the  width  of  the  room 
in  yards  by  the  width  of  the  carpet  in  yards. 

(b)  When  laid  crosswise.  —  Divide  the  length  of  the  room 
in  yards  by  the  width  of  the  carpet  in  yards. 

(2)  To  obtain  the  number  of  yards  of  carpet  needed  to  car- 
pet a  room. 

Rule.  —  Multiply  the  length  of  a  strip  in  yards  {-\- the  fraction 
of  a  yard  allowed  for  waste,  when  considered)  by  the  number  of 
strips. 

Casks  and  Barrels 
To  find  the  contents  in  gallons. 

Rule. — Add  to  the  head  diameter  (inside)  two  thirds  of  the 
difference  between  the  head  and  b^ing  diameters ;  but  if  the  staves 
are  only  slightly  curved,  add  six  tenths  of  this  difference ;  this 
gives  the  mean  diameter  ;  express  it  in  inches,  square  it,  multiply 
it  by  the  length  in  inches  and  this  product  by  .008 J/. :  the  product 
will  be  the  contents  in  liquid  gallons. 


MENSURATION  261 

Circle 

A  circle  is  a  portion  of  a  plane  bounded  by  a  curved  line 
every  point  of  which  is  equally  distant  from  a  point  within 
called  the  center. 

(1)  Formulae. — 
Area  =  irr^,  or  \  ird^. 

Area  =  cPx  .7854,  or  circumference^  X  .07958. 

Circumference  =  diameter  x  3.1416. 

Circumference  =  radius  x  6.2832. 

Diameter  =  circumference  x  .31831. 

Diameter  =  circumference  -^  tt. 

Radius  =  circumference  x  .159155. 

Radius  =  .56419  x  Varea. 

Side  of  inscribed  square  =  dx  .707107. 

Side  of  inscribed  square  =  circumference  x  .22508. 

Area  of  inscribed  square  =  |  cT'. 

Side  of  an  equal  square  =  circumference  X  .282. 

Area  of  an  equal  square  =  f  cP. 

Side  of  inscribed  equilateral  triangle  =  d  x  .86. 

(2)  Given  the  area  inclosed  by  three  equal  circles,  to  find 
the  diameter  of  a  circle  that  will  just  inclose  the  three  equal 
circles. 

Rule.  —  Divide  the  given  area  by  .03473265,  extract  the  square 
root  of  the  quotient,  and  multiply  by  2,  and  the  result  will  be  tlie 
diameter  required. 


Formula.  —  Diameter 


=2\£ 


area 


.03473265 

(3)  To  find  the  diameter  of  the  three  largest  equal  circles 
that  can  be  inscribed  in  a  circle  of  a  given  diameter. 

Rule.  —  Multiply  the  given  diameter  by  .J^G^l,  or  divide  by 
2.1557,  and  the  result  will  be  the  required  diameter. 

D 


Formula.  — d  =  .4641  x  D,  or 


2.1557 


262  MATHEMATICAL   WRINKLES 

(4)  Given  the  radius  a,  b,  c,  of  the  three  circles  tangent  to 
each  other,  to  find  the  radius  of  a  circle  tangent  to  the  three 
circles. 

Formula.  —  r  or  r'  =  — ^^^ ,  the 

2  V[a6c  (a  +  6  +  c)]  T  (ab  -f  ac  -f-  be) 

minus  sign  giving  the  radius  of  a  tangent  circle  circumscribing 
the  three  given  circles,  and  the  plus  sign  giving  the  radius  of  a 
tangent  circle  inclosed  by  the  three  given  circles. 

—  Fi'om  "  The  School  Visitor." 

(5)  Given  the  chord  of  an  arc  and  the  radius  of  the  circle, 
to  find  the  chord  of  half  the  arc. 


Formula.  —  k=  v  2  r^  —  r  V4  r^  —  c^,  where  r  =  radius  and  c  = 
the  given  chord. 

(6)  Given  the  chord  of  an  arc  and  the  radius  of  the  circle, 
to  find  the  height  of  the  arc. 

Rule.  —  From  the  radius,  subtract  the  square  root  of  the  differ- 
ence of  the  squares  of  the  radius  and  half  the  chord. 

Formula.  —  h  =  r  —  Vr^  —  \  c^,  where  r  =  radius  and  c  =  the 
given  chord. 

(7)  Given  the  height  of  an  arc  and  a  chord  of  half  the  arc, 
to  find  the  diameter  of  the  circle. 

Rule.  —  Divide  the  square  of  the  chord  of  half  the  arc  by  the 
height  of  the  chord. 

Formula. — d  =  c^-r-hj  where  c  =  chord  of  half  the  arc  and 
h  =  height. 

(8)  Given  a  chord  and  height  of  the  arc,  to  find  the  chord 
of  half  the  arc. 

Rule.  —  Extract  the  square  root  of  the  sum  of  the  squares  of 
the  height  of  the  arc  and  half  the  chord. 

Formula.  —  A;  =  V^N-T^>  where  h  =  height  and  c  =  the 
given  chord, 


MENSURATION  263 

(9)  Given  the  radius  of  a  circle  and  a  side  of  an  inscribed 
polygon,  to  find  the  side  of  a  similar  circumscribed  polygon. 

2  sr 
Formula.  —  s'  =  ■.  where  s'  =  the  side  required  and 

8  =  the  side  of  the  inscribed  polygon. 

Cone 
A  cone  is  a  solid  bounded  by  a  conical  surface  and  a  plane. 

(1)  To  find  the  lateral  ar6a  of  a  right  circular  cone. 

Rule.  —  Multiply  the  circumference  of  its  base  by  half  the  slant 
height. 

Formula.  —  Lateral  area  =  irrh,  where  r  =  the  radius  of  the 
base  and  h  =  the  slant  height. 

(2)  To  find  the  volume  of  any  cone. 

Rule.  —  Multiply  the  base  by  one  third  the  altitude. 
Formula.  —  V=i  aB. 

Formula,  when  base  is  a  circle.  —  F=  ^  ar^Tr,  where  a  =  alti- 
tude, B  =  base,  and  r  =  radius  of  the  base. 

Crescent 

A  crescent  is  a  portion  of  a  plane  included  between  the  cor- 
responding arcs  of  two  intersecting  circles,  and  is  the  difference 
between  two  segments  having  a  common  chord,  and  on  the 
same  side  of  it. 

Cube  or  Hexahedron 


Diagonal  =  V3  x  edge',  or  Varea  -h  2. 
Diagonal  =  edge  x  1.7320508. 
Surface    =  6  x  edge*,  or  2  x  diagonal*. 
Volume    =  edge'. 

Cycloid 

A  cycloid  is  the  curve  generated  by  a  point  in  the  circum- 
ference of  a  circle  which  rolls  on  a  straight  line. 


264  MATHEMATICAL    WRINKLES 

(1)  To  find  the  length  of  a  cycloid. 

Rule.  —  Multiply  the  diameter  of  the  generating  circle  by  4- 

(2)  To  find  the  area  of  a  cycloid. 

Rule.  —  Multiply  the  area  of  the  generating  circle  by  3. 

(3)  To  find  the  surface  generated  by  the  revolution  of  a 
cycloid  about  its  base. 

Rule.  —  Multiply  the  area  of  the  generating  circle  by  -^^. 

(4)  To  find  the  volume  of  the  solid  formed  by  revolving  the 
cycloid  about  its  base. 

Rule.  —  Multiply  the  cube  of  the  radius  of  the  generating  circle 
bydir", 

(5)  To  find  the  surface  generated  by  revolving  the  cycloid 
about  its  axis. 

Rule.  —  Midtiply  eight  times  the  area  of  the  generating  circle  by 
tr  minus  |. 

(6)  To  find  the  volume  of  the  solid  formed  by  revolving  the 
cycloid  about  its  axis. 

Rule.  —  Multiply  |  of  the  volume  of  a  sphere  whose  radius  is 
that  of  the  generating  circle  by  ^  ir^  —  |. 

(7)  To  find  the  surface  formed  by  revolving  the  cycloid 
about  a  tangent  at  the  vertex. 

Rule.  —  Multiply  the  area  of  the  generating  circle  by  ^^. 

(8)  To  find  the  volume  formed  by  revolving  a  cycloid  about 
a  tangent  at  the  vertex. 

Rule.  —  Multiply  the  cube  of  the  radius  of  the  generating  circle 
byl-r^. 

Cylinder 

A  cylinder  is  a  solid  bounded  by  a  cylindric  surface  and  two 
parallel  planes. 

(1)  To  find  the  lateral  area  of  a  right  circular  cylinder. 
Rule.  —  Multiply  its  length  by  the  circumference  of  its  base. 


MENSURATION  266 

(2)  To  find  the  volume  of  any  cylinder. 

Rule.  —  Multiply  the  altitude  of  the  cylinder  by  the  area  of  its 
base. 

Formula.  —  V=  a  x  B. 

Formula  when  base  is  a  circle.  —  V=  airi^. 

(3)  To  find  the  surface  common  to  two  equal  circular  cylin- 
ders whose  axes  intersect  at  right  angles. 

Rule.  —  Multiply  the  square  of  the  radius  of  the  intersecting 
cylinders  by  16. 

(4)  To  find  the  volume  common  to  two  equal  circular  cylin- 
ders whose  axes  intersect  at  right  angles. 

Rule.  —  Multiply  the  cube  of  the  radius  of  the  intersecting  cylin- 
ders by  iy\. 

(5)  To  find  the  length  of  the  maximum  cylinder  inscribed 
in  a  cube,  the  axis  of  the  cylinder  coinciding  with  the  diagonal 
of  the  cube. 

Formula.  —  Length  =  ^aV3,  where  a  is  the  edge  of  the 
cube. 

(6)  To  find  the  volume  of  the  maximum  cylinder  inscribed 
in  a  cube,  the  axis  of  the  cylinder  coinciding  with  the  diagonal 
of  the  cube. 

Formula. —  F=^?ra^V3,  where  a  is  the  edge  of  the  cube. 

Density  of  a  Body 

The  density  of  any  substance  is  the  number  of  times  the 
weight  of  the  substance  contains  the  weight  of  an  equal  bulk 
of  water. 

To  find  the  density  of  a  body. 

Rule.  —  Divide  the  weight  in  grams  by  the  bulk  in  cubic  cen- 
timeters. 


266  MATHEMATICAL  WBINKLES 

DODECAEDRON 

A  dodecaedron  is  a  polyedron  of  twelve  faces. 

(1)  To  find  the  area  of  a  regular  dodecaedron. 
Rule.  —  Multiply  the  square  of  an  edge  by  20.6^578. 

(2)  To  find  the  volume  of  a  regular  dodecaedron. 
Rule.  —  Multiply  the  cube  of  an  edge  by  7.66312. 

Ellipse 

An  ellipse  is  a  plane  curve  of  such  a  form  that  if  from  any 
point  in  it  two  straight  lines  be  drawn  to  two  given  fixed  points, 
the  sum  of  these  straight  lines  will  always  be  the  same. 

(1)  To  find  the  circumference  of  an  ellipse,  the  transverse 
and  conjugate  diameters  being  known. 

Rule.  —  Multiply  the  square  root  of  half  the  sum  of  the  squares 
of  the  two  diameters  by  3.141592. 

(2)  To  find  the  area  of  an  ellipse,  the  transverse  and  con- 
jugate diameters  being  given. 

Rule.  —  Multiply  the  product  of  the  diameters  by  .785398. 

Frustum  of  a  Cone  or  Pyramid 

A  frustum  of  a  cone  or  pyramid  is  the  portion  included  be- 
tween the  base  and  a  parallel  section. 

(1)  To  find  the  lateral  surface. 

Rule.  —  Midtiply  the  sum  of  the  perimeters,  or  circumferences, 
by  one  half  the  slant  height. 

(2)  To  find  the  entire  surface. 

Rule.  —  Add  to  the  lateral  surface  the  areas  of  both  ends,  or  bases. 

(3)  To  find  the  volume  of  a  frustum  of  a  cone  or  pyramid. 
Rule.  —  To  the  sum  of  the  areas  of  both  bases  add  the  square 

root  of  the  product,  and  midtiply  this  sum  by  07ie  third  of  the 
altitude. 


MENSURATION  267 

Grain  and  Hay 

(1)  To  find  the  quantity  of  grain  in  a  bin. 

Rule.  —  Multiply  the  contents  in  cubic  feet  by  .83 j  and  the  result 
will  be  the  contents  in  bushels. 

(2)  To  find  the  quantity  of  corn  in  a  wagon  bed  or  in  a  ^in. 
Rule.  —  (i)   For  shelled  corny  mxdtiply  the  contents  in  cubic  feet 

by  .83,  and  the  result  will  be  the  contents  in  bushels.  Rule.  —  (S) 
For  com  on  the  cob,  deduct  one  half  for  cob.  Rule.  —  (3)  For  com 
,not  ^^  shucked  "  deduct  two  thirds  for  cob  and  shuck. 

(3)  To  find  the  quantity  of  hay  in  a  stack  or  rick. 

Rule.  — Divide  the  contents  in  cubic  feet  by  550  for  clover  or  by 
450  for  timothy;  the  quotient  will  be  the  number  of  tons. 

(4)  In  well-settled  stacks  15  cubic  yards  make  one  ton. 

(5)  When  hay  is  baled,  10  cubic  yards  make  one  ton. 

Hexaedron 
(See  Cube.) 

Hyperbola 

A  hyperbola  is  a  section  formed  by  passing  a  plane  through 
a  cone  in  a  direction  to  make  an  angle  at  the  base  greater  than 
that  made  by  the  slant  height. 

To  find  the  area  of  a  hyperbola,  the  transverse  and  conjugate 
axes  and  abscissa  being  given. 

Rule.  —  (1)  To  the  product  of  the  transverse  diameter  and 
absci-fsa  add  ^  of  the  square  of  the  abscissa,  and  multiply  the 
square  root  of  the  sum  by  21. 

(2)  Add  4  times  the  square  root  of  the  product  of  the  trans- 
verse diameter  and  abscissa  to  the  product  last  found,  and  divide 
the  sum  by  75. 

(3)  Divide  4  times  the  product  of  the  conjugate  diameter  and 
abscissa  by  the  transverse  diameter,  and  this  last  quotient  multi- 
plied by  the  former  will  give  the  area  required^  nearly. 


268  MATHEMATICAL   WRINKLES 

ICOSAEDRON 

An  icosaedron  is  a  polyedron  of  twenty  faces. 

(1)  To  find  the  area  of  a  regular  icosaedron. 
Rule.  —  Multiply  the  square  of  an  edge  by  8.66025. 

(2)  To  find  the  volume  of  a  regular  icosaedron. 
Rule.  —  Multiply  the  cube  of  an  edge  by  2.18169. 

Irregular  Polyedron 
To  find  the  volume  of  any  irregular  polyedron. 
Rule.  —  Cut  the  polyedron  into  prismatoids  by  passing  parallel 
planes  through  all  its  summits. 

Irregular  Solids 
To  find  the  volume  of  any  irregular  solid. 
Rule.  —  Immerse  the  solid  in  a  vessel  of  water  and  determine  the 
quantity  of  water  displaced. 

Logs 

(1)  To  find  the  side  of  the  squared  timber  that  can  be  sawed 
from  a  log. 

Rule. — Multiply  the  diameter  of  the  smaller  end  by  .707. 

(2)  To  find  the  number  of  board  feet  in  the  squared  timber 
that  can  be  sawed  from  a  log. 

Rule.  —  Multiply  together  one  half  the  length  in  feet,  the  diameter 
of  the  smaller  end  in  feet,  and  the  diameter  of  the  smaller  end  in 
inches. 

Problem.  —  Find  the  side,  and  the  number  of  board  feet,  in 
the  squared  timber  that  can  be  sawed  from  a  log  whose  length 
is  16  feet,  and  diameter  of  the  smallest  end  15  inches. 

Solution.  — By  (1)  the  side  is  15  inches  x  .707,  or  10.606  inches. 
By  (2)  the  number  of  the  board  feet  is  \^-  x  ^f  x  15  =  150,  Ans. 


MENSURATION  269 

Lumber 

When  boards  are  1  inch  thick  or  less,  they  are  estimated  by 
the  square  foot  of  surface,  the  thickness  not  being  considered. 

Thus  a  board  10  feet  long,  1  foot  wide,  and  1  inch  (or  less) 
thick  contains  10  square  feet. 

Hence,  to  find  the  number  of  board  feet  in  a  plank. 

Rule.  —  Multiply  the  length  in  feet  by  the  width  in  feet  by  the 
thickness  in  inches. 

Note. — The  average  width  of  a  board  that  tapers  uniformly  is  one 
half  the  sum  of  the  end  widths. 

LUNE 

A  lune  is  that  portion  of  a  sphere  comprised  between  two 
great  semicircles. 

To  find  the  area  of  a  lune. 

Rule.  —  Multiply  its  angle  in  radians  by  twice  the  square 
of  the  radius. 

OCTAEDRON 

An  octaedron  is  a  polyedron  of  eight  faces. 

(1)  To  find  the  area  of  a  regular  octaedron. 
Rule.  —  Multiply  the  square  of  an  edge  by  S.j^G^I' 

(2)  To  find  the  volume  of  a  regular  octaedron. 
Rule.  —  Multiply  the  cube  of  an  edge  by  .4714- 

Painting  and  Plastering 

Painting  and  plastering  are  usually  estimated  by  the  square 
yard.  The  processes  of  calculating  the  cost  of  painting  and 
plastering  vary  so  much  in  different  localities  that  it  is  impos- 
sible to  lay  down  any  rule.  Usually  some  allowance  is  made 
for  doors,  windows,  etc.,  but  there  is  no  fixed  rule  as  to  how 
much  should  be  deducted.  Sometimes  one  half  the  area  of  the 
openings  is  deducted. 


270  MATHEMATICAL   WRINKLES 

Papering 

Wall  paper  is  sold  only  by  the  roll,  and  any  part  of  a  roll  is 
considered  a  whole  roll. 

The  amount  of  wall  paper  required  to  paper  a  room  depends 
upon  the  area  of  the  walls  and  ceiling  and  the  waste  in  matching. 

(1)  American  paper  is  commonly  18  inches  wide,  and  has  8 
yards  in  a  single  roll,  and  16  yards  in  a  double  roll.  Foreign 
papers  vary  in  width  and  length  to  the  roll. 

(2)  Wall  paper  is  usually  put  up  in  double  rolls,  but  the 
prices  quoted  are  for  single  rolls. 

(3)  Borders  and  friezes  are  sold  by  the  yard  and  vary  in 
width. 

(4)  The  area  of  a  single  roll  is  36  square  feet,  and  allowing 
for  all  waste  in  matching,  etc.,  will  cover  30  square  feet  of  wall. 

(5)  There  is  no  fixed  rule  as  to  how  much  should  be  deducted 
for  doors  and  windows.  Some  dealers  deduct  the  exact  area  of 
the  openings,  while  others  deduct  an  approximate  area,  allowing 
20  square  feet  for  each. 

(6)  The  number  of  single  rolls  required  for  the  ceiling  and 
for  the  walls  must  be  estimated  separately. 

(7)  To  obtain  the  number  of  single  rolls  required  for  the 
ceiling. 

Rule.  —  Divide  its  area  in  square  feet  by  30. 

(8)  To  obtain  the  number  of  single  rolls  required  for  the 
walls. 

Rule.  —  From  the  area  of  the  walls  in  square  feet  deduct  the 
area  of  the  openings,  and  divide  by  30. 

Parabola 

A  parabola  is  the  locus  of  a  point  whose  distance  from  a 
fixed  point  is  always  equal  to  its  distance  from  a  fixed  straight 
line. 


MENSURATION  271 

(1)  To  find  the  length  of  any  arc  of  a  parabola  cut  off  by  a 
double  ordinate. 

Rule.  —  When  the  abscissa  is  less  than  half  the  ordinate :  To 
t/ie  square  of  the  ordinate  add  J  of  the  square  of  the  abscissa,  and 
twice  the  square  root  of  the  sum  will  be  the  length  of  the  arc. 

(2)  To  find  the  area  of  the  parabola,  the  base  and  height 
being  given. 

Rule.  —  Multiply  the  bcLse  by  the  heighty  and  ^  of  the  product 
will  be  the  area. 

(3)  To  find  the  area  of  a  parabolic  frustum,  having  given 
the  double  ordinates  of  its  ends  and  the  distance  between  them. 

Rule.  —  Divide  the  difference  of  the  cubes  of  the  two  ends  by  the 
difference  of  their  squares  and  multiply  tlie  quotient  by  J  of  the 
altitude. 

Parallelogram 

A  parallelogram  is  a  quadrilateral  whose  opposite  sides  are 
parallel. 

To  find  the  area  of  any  parallelogram. 
Rule.  —  Multiply  the  base  by  the  altitude. 

Parallelopiped 
A  parallelopiped  is  a  prism  whose  bases  are  parallelograms. 
To  find  the  volume  of  any  parallelopiped. 
Rule.  —  Multiply  its  altitude  by  the  area  of  its  base. 

Prism 

A  prism  is  a  polyedron  whose  ends  are  equal  and  parallel 
polygons,  and  its  sides  parallelograms. 

(1)  To  find  the  lateral  area  of  a  prism. 

Rule.  —  Multiply  a  lateral  edc/e  by  the  perimeter  of  a  right  sec- 
tion. 


272  .      MATHEMATICAL   WEINKLES 

(2)  To  find  the  volume  of  any  prism. 

Rule.  —  Multiply  the  area  of  the  base  by  its  altitude. 

Prismatoid 

A  prismatoid  is  a  polyedron  whose  bases  are  any  two  poly- 
gons in  parallel  planes,  and  whose  lateral  forces  are  triangles 
determined  by  so  joining  the  vertices  of  these  bases  that  each 
lateral  edge  with  the  preceding  forms  a  triangle  with  one  side 
of  either  base. 

(1)  To  find  the  volume  of  any  prismatoid. 

Rule.  —  Add  the  areas  of  the  two  bases  and  four  times  the  mid 
c7^oss  section;  multiply  this  sum  by  one  sixth  the  altitude. 
Old  Prismoidal  Formula.  — 

(2)  To  find  the  volume  of  a  prismatoid,  or  of  any  solid  whose 
section  gives  a  quadratic. 

Rule.  —  Multiply  one  fourth  its  altitude  by  the  sum  of  one  ba^e 
and  three  times  a  section  distant  from  that  base  two  thirds  the 
altitude. 

New  Prismoidal  Formula.  — 

V=-(B  +  3T). 
—  From  Halsted's  "Metrical  Geometry." 

Pyramid 

A  pyramid  is  a  polyedron  of  which  all  the  faces  except  one 
meet  in  a  point. 

(1)  To  find  the  lateral  area  of  a  regular  pyramid. 

Rule.  —  Multi2)ly  the  perimeter  of  the  base  by  half  the  slant 
height. 

(2)  To  find  the  volume  of  any  pyramid. 

Rule.  —  Multiply  the  area  of  the  base  by  one  third  of  the  altitude. 


MENSURATION  273 

Pyramid,  Spherical 

A  spherical  pyramid  is  the  portion  of  a  sphere  bounded  by 
a  spherical  polygon  and  the  planes  of  its  sides. 

Rule.  —  Multiply  the  area  of  the  hose  by  one  third  of  the  radius 
of  the  sphere. 

Note.  —  The  area  of  a  spherical  polygon  Is  equivalent  to  a  lune  whose 
angle  is  half  the  spherical  excess  .of  the  polygon. 

Quadrilateral 
A  quadrilateral  is  a  polygon  of  four  sides. 
To  find  the  area  of  any  quadrilateral. 

Rule.  —  Multiply  half  the  diagonal  by  the  sum  of  the  perpen- 
diculars upon  it  from  the  opposite  angle. 

Rhombus 
A  rhombus  is  a  parallelogram  whose  sides  are  all  equal  and 
whose  angles  are  oblique. 

To  find  the  area  of  a  rhombus. 

Rule.  —  Take  half  the  product  of  Us  diagonals. 

Rings 

If  a  plane  curve  lying  wholly  on  the  same  side  of  a  line  in  its 
own  plane  revolves  about  that  line,  the  solid  thus  generated  is 
called  a  ring. 

(1)  Theorem  of  Pappus. 

(a)  If  a  plane  curve  lying  wholly  on  the  same  side  of  a  line 
in  its  own  plane  revolves  about  that  line,  the  area  of  the  solid 
thus  generated  is  equal  to  the  product  of  the  length'  of  the  re- 
volving line  and  the  path  described  by  its  center  of  mass. 

(6)  If  a  plane  figure  lying  wholly  on  the  same  side  of  a  line 
in  its  own  plane  revolves  about  that  line,  the  volume  of  the 
solid  thus  generated  is  equal  to  the  product  of  the  revolving 
area  and  the  length  of  the  path  described  by  its  center  of  mass. 


274  MATHEMATICAL   WRINKLES 

(2)  To  find  the  surface  of  an  elliptic  ring. 
Formula.  —  Surface  =  2  tt^  c  V|((2a)^  +  (26)2). 

(3)  To  find  the  volume  of  an  elliptic  ring. 

Formula.  —  Volume  =  2  -n-^abc,  where  2  a  and  2  b  are  the  axes 
of  the  ellipse  and  c  the  distance  of  the  center  of  the  ellipse 
from  the  axis  of  rotation. 

(4)  To  find  the  surface  of  a  cylindric  ring. 
Formula.  —  Surface  =  4  ii^ra. 

(5)  To  find  the  volume  of  a  cylindric  ring. 

Formula.  — Volume  =  27rVa,  where  a  =  distance  of  the  center 
of  the  generating  curve  from  the  axis  of  rotation,  and  r  =  the 
radius  of  the  circle. 

Roofing  and  Flooring 

A  square  10  feet  on  a  side,  or  100  square  feet,  is  the  unit  of 
measure  in  roofing,  tiling,  and  flooring. 

The  average  shingle  is  taken  to  be  16  inches  long  and  4  inches 
wide.     Shingles  are  usually  laid  about  4  inches  to  the  weather. 

When  laid  4^  inches  to  the  weather,  the  exposed  surface  of 
a  shingle  is  18  square  inches. 

Allowing  for  waste,  about  1000  shingles  are  estimated  as 
needed  for  each  square,  but  if  the  shingles  are  good,  850  to  900 
are  sufficient.     There  are  250  shingles  in  a  bundle. 

Sector 

A  sector  is  that  portion  of  a  circle  bounded  by  two  radii  and 
the  intercepted  arc. 

To  find  the  area  of  a  sector. 

Rule.  —  (a)  Multiply  the  length  of  the  arc  by  half  the  radius. 
(b)  If  the  arc  is  given  in  degrees,  take  such  a  part  of  the  whole 
area  of  the  circle  as  the  number  of  degrees  in  the  arc  is  of  360°. 


MENSURATION  275 

Sector,  A  Spherical 

A  spherical  sector  is  the  volume  generated  by  any  sector  of 
a  semicircle  which  is  revolved  about  its  diameter. 

To  find  the  volume  of  a  spherical  sector. 
Rule.  —  Multiply  the  area  of  its  zone  by  one  third  the  radius. 
Formula.  —  V=  |  irar^y  where  r  =  radius  of  the  sphere  and 
a  =  altitude  of  the  spherical  segment. 

Segment  of  Circle 

A  segment  of  a  circle  is  the  portion  of  a  circle  included  be- 
tween an  arc  and  its  chord. 

(1)  To  find  the  area  of  a  segment  less  than  a  semicircle. 

Rule. — From  the  sector  having  the  same  arc  a.9  the  segment 
subtract  the  area  of  the  triangle  formed  by  the  chord  and  the  two 
radii  from  its  extremities. 

(2)  An  approximate  rule  for  finding  the  area  of  a  segment. 
Rule. —  Take  two  thirds  the  product  of  its  chord  and  height. 

(3)  To  find  the  area  of  a  segment  of  a  circle,  having  given 
the  chord  of  the  arc  and  the  height  of  the  segment,  i.e.  the 
versed  sine  of  half  the  arc. 

Rule.  —  Divide  the  cube  of  the  height  by  twice  the  base  and  in- 
crease the  quotient  by  two  thirds  of  the  product  of  the  height  and 
base. 

(4)  To  find  the  volume  of  the  solid  generated  by  a  circular 
segment  revolving  about  a  diameter  exterior  to  it. 

Rule.  — Multii)ly  one  sixth  the  area  of  the  circle  tvhose  radius 
is  the  chord  of  the  segment  by  the  projection  of  that  chord  upon 
the  axis. 

Formula.  —  F=  ;  TT^LB'-^  X  ^I'B',  where  AB  is  the  chord  of 
the  segment  and  A'B'  is  its  projection  upon  the  axis. 


276  MATHEMATICAL   WRINKLES 

Segment,  A  Spherical 

A  spherical  segment  is  a  portion  of  a  sphere  contained  be- 
tween two  parallel  planes. 

To  find  the  volume  of  any  spherical  segment. 

Rule.  —  To  the  product  of  one  half  the  sum  of  its  bases  by  its 
altitude  add  the  volume  of  a  sphere  having  that  altitude  for  its 
diameter. 

Shell,  A  Cylindric 

A  cylindric  shell  is  the  difference  between  two  circular 
cylinders  of  the  same  length. 

To  find  the  volume  of  a  cylindric  shell. 

Rule.  —  Multiply  the  sum  of  the  inner  and  outer  radii  by  their 
difference,  and  this  product  by  ir  times  the  altitude  of  the  shell. 

Shell,  A  Spherical 

A  spherical  shell  is  the  difference  between  two  spheres  which 
have  the  same  center. 

To  find  the  volume  of  a  spherical  shell. 

Formula.  —  V=  f  ir  (r^^  —  r^),  where  rj  and  r  denote  the 
radii. 

Similar  Solids 

Similar  solids  are  solids  which  have  the  same  form,  and  dif- 
fer from  each  other  only  in  volume. 

Rule.  —  Any  two  similar  solids  are  to  each  other  as  the  cubes 
of  any  two  like  dimensions. 

Similar  Surfaces 

Similar  surfaces  are  surfaces  which  have  the  same  shape, 
and  differ  from  each  other  only  in  size. 

Rule. — Any  two  similar  surfaces  are  to  each  other  as  the  squares 
of  any  two  like  dimensions. 


MENSURATION  277 

Sphere 
A  sphere  is  a  closed  surface  all  points  of  which  are  equally 
distant  from  a  fixed  point  within  called  the  center. 
(1)   Formulae. — 

Area  =  4  ttt^,  or  vd^. 
Area  =  7^x12.5664. 
Area  =  d2x  3.1416. 
Area  =  circumference'  x  .3183. 
Volume  =  I  nr^f  or  J  7rd\ 
Volume  =  J  d  X  area. 
Volume  =  circumference''  x  .0169. 
Volume  =  r«  x  4.1888,  or  d»  x  .5236. 


(2)   Side  of  an  inscribed  cube 


(  r  X  1.1547, 

=  -J  or 

(  d  X  .5774. 


(3)  To  find  the  edge  of  the  largest  cube  that  can  be  cut  from 
a  hemisphere. 

Formula.  — Edge  =d  x  .408248. 

(4)  To  find  the  volume  of  a  frustum  of  a  sphere,  or  the  por- 
tion included  between  two  parallel  planes. 

Rule.  —  To  three  times  the  sum  of  the  squared  radii  of  the  two 
ends  add  the  square  of  the  altitude  ;  multiply  this  sum  by  .5235987 
times  the  altitude. 

(5)  To  find  the  edge  of  the  largest  cube  that  can  be  inscribed 
in  a  hemisphere  of  given  diameter. 

Rule.  —  Multiply  the  radius  by  ^  of  the  square  root  of  6, 

Spheroid 
A  spheroid  is  a  solid  formed  by  revolving  an  ellipse  about 
one  of  its  axes  as  an  axis  of  revolution. 

Spheroid,  Oblate 

An  oblate  spheroid  is  the  spheroid  formed  by  revolving  an 
ellipse  about  its  conjugate  diameter  as  an  axis  of  revolution.   ^ 


278  MATHEMATICAL   WRINKLES 

To  find  the  volume  of  an  oblate  spheroid. 
Rule.  —  Multiply  the  square  of  the  semitransverse  diameter  by 
the  semiconjugate  diameter  and  this  product  by  ^  tt. 

Spheroid,  Prolate 

A  prolate  spheroid  is  the  spheroid  formed  by  revolving  an 
ellipse  about  its  transverse  diameter  as  an  axis  of  revolution. 

To  find  the  volume  of  a  prolate  spheroid. 
Rule.  —  Multiply  the  square  of  the  semiconjugate  diameter  by 
the  semitransverse  diameter  and  this  product  by  ^  ir. 

Spindle,  A  Circular 

A  circular  spindle  is  the  solid  formed  by  revolving  the  seg- 
ment of  a  circle  about  its  chord. 

(1)  To  find  the  volume  of  a  circular  spindle. 

Rule.  —  Multiply  the  area  of  the  generating  segment  by  the  path 
of  its  center  of  gravity. 

(2)  To  find   the  volume  formed  by  revolving  a  semicircle 
about  a  tangent  parallel  to  its  diameter. 

Rule.  —  Multiply  one  fourth  of  the  volume  of  a  sphere  ivhose 
radius  is  that  of  the  generating  semicircle  by  (10  —  S  ir). 

Spindle,  A  Parabolic 

A  parabolic  spindle  is  a  solid  formed  by  revolving  a  parabola 
about  a  double  ordinate  perpendicular  to  the  axis. 

To  find  the  volume  of  a  parabolic  spindle. 

Rule.  —  Multiply  the  volume  of  its  circumscribed  cylinder  by  ^. 

Square 
A  square  is  a  rectangle  whose  sides  are  all  equal, 
(1)   To  find  the  area  of  a  square. 
Rule.  —  /Square  an  edge. 


MENSURATION  279 

(2)  Given  the  diagonal,  to  find  the  area. 
Rule.  —  Take  one  half  the  square  of  the  diagonal. 

(3)  Given  the  diagonal,  to  find  a  side. 

Rule.  —  Extract  the  square  root  of  one  half  the  square  of  the 
diayonal. 

(4)  To  find  the  side  of  the  largest  square  inscribed  in  a 
semicircle  of  given  diameter. 

Rule.  —  Multiply  the  radius  of  the  given  circle  by  |  of  the 
square  root  of  5. 

Tetraedron 

A  tetraedron  is  a  polyedron  of  four  faces. 

(1)  To  find  the  surface  of  a  tetraedron. 

Rule.  —  Mtdtiply  the  square  of  an  edge  by  V^,  or  1.73205. 

(2)  To  find  the  volume  of  a  tetraedron. 

Rule.  —  Multiply  the  cube  of  an  edge  by  ■^'^2^  or  .11785. 

Trapezium  and  Irregular  Polygons 
To  find  the  area  of  a  trapezium  or  any  irregular  polygon. 
Rule.  —  Divide  the  figure  into  triangles^  find  the  area  of  the 
triangles,  and  take  their  sum. 

Trapezoid 

A  trapezoid  is  a  quadrilateral  two  of  whose  sides  are  par- 
allel. 

(1)  To  find  the  area  of  a  trapezoid. 

Rule.  —  Multiply  the  altitude  by  one  half  the  sum  of  the  parallel 
sides. 

(2)  Width  =  area  -t-  (^  of  the  sum  of  the  parallel  sides). 

(3)  Sura  of  the  parallel  sides  =  (area  -^  width)  x  2. 

(4)  To  find  the  length  of  a  line  parallel  to  the  bases  of  a 
trapezoid  that  shall  divide  it  into  equal  areas. 


280  MATHEMATICAL  WRINKLES 

Rule.  —  Square  the  bases  and  extract  the  square  root  of  half 
their  sum. 

Triangle 

A  triangle  is  a  portion  of  a  plane  bounded  by  three  straight 
lines. 

(1)  To  find  the  area  of  a  triangle. 

Rule.  —  Multiply  the  base  by  half  the  altitude. 

(2)  To  find  the  area  of  a  triangle,  having  given  the  three 
sides. 

Rule.  —  From  half  the  sum  of  the  three  sides  subtract  each  side 
separately;  multiply  half  the  sum  and  the  three  remainders  to- 
gether :  the  square  root  of  the  product  will  be  the  area. 

(3)  To  find  the  radius  of  the  inscribed  circle. 

Rule.  —  Divide  the  area  of  the  triangle  by  half  the  sum  of  its 
sides. 

(4)  To  find  the  radius  of  the  circumscribing  circle. 

Rule.  —  Divide  the  product  of  the  three  sides  by  four  times  the 
area  of  the  triangle. 

(5)  To  find  the  radius  of  an  escribed  circle. 

Rule.  —  Divide  the  area  of  the  triangle  by  the  difference  between 
half  the  sum  of  its  sides  and  the  tangent  side. 

(6)  To  cut  off  a  triangle  containing  a  given  area  by  a  line 
running  parallel  to  one  of  its  sides,  having  given  the  area  and 
base. 

Rule. —  The  area  of  the  given  triangle  is  to  the  area  of  the  tri- 
angle to  be  cut  off,  as  the  square  of  the  given  base  is  to  the  square 
of  the  required  base.     Extract  the  square  root  of  the  result. 

(Equilateral)  Triangle 

(1)  Area  =  one  half  the  side  squared  and  multiplied  by  V3, 
or  1.732050+. 


MENSURATION  281 

(2)  Altitude  =  one  half  the    side    multiplied    by    V3,   or 
1.732050^. 

(3)  Center  of  the  inscribed  and  circumscribed  circle  is  a 
point  in  the  altitude  one  third  of  its  length  from  the  base. 

(4)  Radius  of  the  circumscribed  circle  =  two  thirds  of  the 
altitude. 

(5)  Radius  of  the  inscribed  circle  =  one  third  of  the  altitude. 


(6)  Side  =  2  Varea^W3. 

Side  =  radius  of  the  circumscribed  circle  multiplied  by 

vs. 

(7)  All  equilateral  triangles  are  similar. 

(8)  Each  angle  =  60°. 

(Right)  Triangle 

(1)  B&ae^y/ih'-f). 

(2)  Perpendicular  =  V(/i*  -  6*). 


(3)  Hypotenuse  =V6N-p. 

(4)  Diameter  of  inscribed  circle  =  (b+p)  —  h. 

(6)  Side  opposite  an  acute  angle  of  30°  =  one  half  of  the 
hypotenuse. 

(6)  Similar,  if  an  acute  angle  of  one  =  an  acute  angle  of 
another. 

(7)  Altitude  of  an  isosceles  triangle  forms  two  right  triangles. 

(8)  To  find  a  point  in  a  right-angled  triangle  equidistant 
from  its  vertices. 

Rule.  —  Divide  the  hypotenuse  by  2;  the  point  will  lie  in  the 
hypotenuse. 

(9)  To  find  the  perpendicular  height  of  a  right  triangle  when 
the  base  and  the  sum  of  the  perpendicular  and  hypotenuse  are 
known. 


282  MATHEMATICAL   WRINKLES 

Rule. — From  the  square  of  the  sum  of  the  perpendicular  and 
hypotenuse  take  the  square  of  the  base,  and  divide  the  difference 
by  twice  the  sum  of  the  perpendicular  and  hypotenuse. 

(Spherical)  Triangle 

A  spherical  triangle  is  a  spherical  polygon  of  three  sides. 

To  find  the  area  of  a  spherical  triangle. 

Rule.  —  Find  the  area  of  a  lune  whose  angle  is  half  the  spheri- 
cal excess  of  the  triangle. 

Note.  —  The  spherical  excess  of  a  triangle  is  the  excess  of  the  sum  of 
its  angles  over  180°. 

Ungula,  a  Conical 

A  conical  ungula  is  a  portion  of  a  cone  cut  off  by  a  plane 
oblique  to  the  base  and  contained  between  this  plane  and  the 
base. 

To  find  the  volume  of  a  conical  ungula,  when  the  cutting 
plane  passes  through  the  opposite  extremes  of  the  ends  of  the 
frustum. 

Rule.  —  Multiply  the  difference  of  the  square  roots  of  the  cubes 
of  the  radii  of  the  bases  by  the  square  root  of  the  cube  of  the 
radius  of  the  lower  base  and  this  product  by  ^tt  times  the  altitude. 
Divide  this  last  product  by  the  difference  of  the  radii  of  the  two 
bases,  and  the  quotient  will  be  the  volume  of  the  ungida. 

Ungula,  A  Cylindric 

A  cylindric  ungula  is  any  portion  of  a  cylinder  cut  off  by  a 
plane. 

(1)  To  find  the  convex  surface  of  a  cylindric  ungula,  when 
the  cutting  plane  is  parallel  to  the  axis  of  the  cylinder. 

Rule.  —  Multiply  the  arc  of  the  base  by  the  altitude. 

(2)  To  find  the  volume  of  a  cylindric  ungula  whose  cutting 
plane  is  parallel  to  the  axis. 


MENSURATION  283 

Rule.  —  Multiply  the  area  of  the  base  by  the  altitude. 

(3)  To  find  the  convex  surface  of  a  cylindric  ungula,  when 
the  plane  passes  obliquely  through  the  opposite  sides  of  the 
cylinder. 

Rule.  —  Multiply  the  circumference  of  the  base  by  half  the  sum 
of  the  greatest  and  least  lengths  of  the  ungula. 

(4)  To  find  the  volume  of  a  cylindric  ungula,  when  the  plane 
passes  obliquely  through  the  opposite  sides  of  the  cylinder. 

Rule.  —  Multiply  the  area  of  the  base  by  half  the  least  and 
greatest  lengths  of  the  ungula. 

Ungula,  A  Spherical 

A  spherical  ungula  is  a  portion  of  a  sphere  bounded  by  a 
lune  and  two  great  semicircles. 

To  find  the  volume  of  a  spherical  ungula. 

Rule.  —  Multiply  the  area  of  the  lune  by  one  third  the  radius; 
OTy  multiply  the  volume  of  the  sphere  by  the  quotient  of  the  angle 
of  the  lune  divided  by  360°. 

Wedge 

A  wedge  is  a  prismatoid  whose  lower  base  is  a  rectangle, 
and  upper  base  a  sect  parallel  to  a  basal  edge. 

To  find  the  volume  of  any  wedge. 

Rule.  —  To  twice  the  length  of  the  ba^e  add  the  opposite  edge; 
mtdtiply  the  sum  by  the  width  of  the  base,  and  this  product  by  one 
sixth  the  altitude  of  the  wedge. 

Wood  Measure 

The  unit  of  wood  measure  is  the  cord.  The  cord  is  a  pile  of 
wood  8  feet  by  4  feet  by  4  feet. 

A  pile  of  wood  1  foot  by  4  feet  by  4  feet  is  called  a  cord 
foot. 


284  MATHEMATICAL   WRINKLES 

A  cord  of  stove  wood  is  8  feet  long  by  4  feet  high.     The 
length  of  stove  wood  is  usually  16  in. 

Zone 

A  zone  is  the  curved  surface  of  a  sphere  included  between 
two  parallel  planes  or  cut  off  by  one  plane. 

(1)  To  find  the  area  of  a  zone. 

Rule.  —  Multiply  the  altitude  of  the  spherical  segment  by  tivice 
IT  times  the  radius  of  the  sphere. 

(2)  To  find  the  area  of  a  zone  of  one  base. 

Rule.  —  Tlie  area  of  a  zone  of  one  base  is  equivalent  to  the  area 
of  a  circle  whose  radius  is  the  chord  of  the  generating  arc. 

(Circular)  Zone 

A  circular  zone  is  the  portion  of  a  plane  inclosed  by  two 
parallel  chords  and  their  intercepted  arcs. 

(1)  If  both  chords  are  on  the  same  side  of  the  center. 
Rule.  —  Find  the  difference  between  the  areas  of  the  tioo  seg- 
ments. 

(2)  If  the  chords  are  on  opposite  sides  of  the  center. 

Rule.  —  Subtract  the  sum  of  the  areas  of  the  two  segments  from 
the  area  of  the  circle. 


MISCELLANEOUS   HELPS 

1.  Pi  (tt)  =  3.1416,  or  3|.  Its  value  to  seven  hundred  and 
seven  places  is 

3.14159265358979323846264338327950288419716939937510582 
09749445923078164062862089986280348253421170679821480 
86513282306647093844609550582231725359408128481117450 
28410270193852110555964462294895493038196442881097566 
59334461284756482337867831652712019091456485669234603 
48610454326648213393607260249141273724587006606315588 
17488152002096282925409171536436789259036001133053054 
88204665213841469519415116094330572703657595919530921 
86117381932611793105118548074462379834749567351885762 
72489122793818301194912983367336244193664308602139501 
60924480772309430285530966202755693979869502224749962 
06074970304123668861995110089202383770213141694119029 
88582544681639799904659700081700296312377381342084130 
791451183980570985. 

2.  The  contents  of  a  spheroid  equals  the  square  of  the  re- 
volving axis  X  the  fixed  axis  x  .5236. 

3.  To  find  the  distance  a  spot  on  the  tire  of  a  revolving 
wheel  moves,  multiply  the  distance  traveled  by  4  and  divide 
by  TT. 

4.  Sound  travels  1087  feet  per  second  at  0**  C.  or  1126  feet 
per  second  at  20°  C. 

6.  Electricity  travels  about  186,000  miles  per  second. 

6.  To  find  the  approximate  number  of  bushels  of  corn  in  a 
crib,  take  the  dimensions  in  feet,  and  multiply  their  product 

286 


286  MATHEMATICAL   WEINKLES 

by  .8,  if  the  corn  is  shelled ;  by  A,  if  shucked ;  by  .3,  if  in  the 
shuck. 

7.  Eoofing,  flooring,  and  slating  are  often  estimated  by  the 
square,  which  contains  100  square  feet. 

8.  The  long  ton  of  2240  pounds  and  the  long  hundredweight 
of  112  pounds  are  used  in  the  United  States  custom  houses 
and  in  weighing  coal  and  iron  in  the  mines. 

9.  The  term  carat  is  sometimes  used  to  express  the  fineness 
of  gold,  each  carat  meaning  a  twenty -fourth  part. 

10.  It  takes  1000  shingles  to  cover  100  square  feet  laid  4 
inches  to  the  weather.  It  takes  900  shingles  to  cover  100  square 
feet  laid  4|-  inches  to  the  weather. 

11.  The  area  of  an  ellipse  is  a  mean  proportional  between  the 
circumscribed  and  inscribed  circles. 

12.  Gunter's  chain  is  66  feet  long,  consisting  of  100  links. 

13.  The  first  24  periods  of  numeration  are — units,  thousands, 
millions,  billions,  trillions,  quadrillions,  quintillions,  sextillions, 
septillions,  octillions,  nonillions,  decillions,  undecillions,  duode- 
cillions,  tredecillions,  quartodecillions,  quintodecillions,  sexde- 
cillions,  septodecillions,  octodecillions,  nonodecillions,  vigin- 
tillions,  primo-vigintillions,  and  secundo  vigintillions. 

14.  Mathematicians  have  given  the  signs  X  and  -;-  precedence 
over  the  signs  +  and  —  ;  hence  the  operations  of  multiplication 
and  division  should  always  be  performed  before  addition  and 
subtraction. 

15.  The  true  weight  of  an  article  weighed  on  false  scales  is 
a  mean  proportional  between  the  two  apparent  weights. 

16.  To  find  any  term  of  an  arithmetical  progression. 

Rule.  —  Any  ter^n  of  an  arithmetical  series  is  equal  to  the  first 
term,  increased  or  diminished  by  the  common  difference  multiplied 
by  a  number  one  less  than  the  number  of  terms. 


MISCELLANEOUS   HELPS-  287 

17.  To  find  the  sum  of  an  arithmetical  series. 

Rule.  —  Multiply  half  the  sum  of  the  extremes  by  the  number  of 
terms. 

18.  To  find  any  term  of  a  geometrical  series. 

Rule.  —  Multiply  the  first  term  by  the  ratio  raised  to  a  power 
one  less  than  the  number  of  terms. 

19.  To  find  the  sum  of  a  geometrical  series. 

Rule.  —  Multiply  the  greater  extreme  by  the  ratio,  subtract  the 
less  extreme  from  the  product,  and  divide  the  remainder  by  the  ratio 
less  1. 

20.  To  sum  a  geometrical  series  to  infinity. 

Rule.  —  When  the  ratio  is  a  proper  fraction,  divide  the  first  term 
by  1  less  the  ratio. 

21.  To  find  the  harmonic  mean  between  two  numbers. 
Rule.  —  Divide  twice  their  product  by  their  sum. 

22.  To  find  the  mean  proportional  between  two  numbers. 
Rule.  —  Take  the  square  root  of  their  product. 

23.  A  body  immersed  in  a  liquid  is  buoyed  up  by  a  force 
equal  to  the  weight  of  the  liquid  displaced.  That  is,  it  loses  a 
portion  of  its  weight  just  equal  to  the  weight  of  the  water  dis- 
placed. 

24.  If  we  have  the  sum  and  difference  of  two  numbers  given, 
add  the  sum  and  difference  and  take  half  of  it  for  the  greater, 
subtract  and  take  half  of  it  for  the  smaller. 

25.  To  find  the  day  of  the  week  for  any  date. 

Rule.  —  To  the  given  year  of  the  century  add  its  \,  neglecting 
remainder  ;  to  this  add  the  day  of  the  month,  the  raJtio  of  the  cen- 
tury, and  the  ratio  of  the  month;  then  divide  by  7,  and  the  re- 
mainder will  be  the  number  of  the  day  of  tJie  week,  counting 
Sunday  1st,  Monday  2d,  and  so  on. 


Monthly 

Ratios 

January- 

=  3  or  2. 

August        =  5. 

February 

=  6  or  5. 

September  =  1. 

March 

=  6. 

October       =  3. 

April 

=  2. 

November  =  6. 

May 

=  4. 

December  =  1. 

June 

=  0. 

In  leap  years 

July 

=  2. 

Jan.  =  2.  reb.:= 

288  MATHEMATICAL   WRINKLES 

Centennial  Ratio 
200,  900,  1800,  2200  =  0. 
300,  1000  .  .  .  .  =  6. 
400,  1100,  1900,  2300  =  5. 
500,  1200,  1600,  2000  =  4. 
600,  1300  .  .  .  .  =  3. 
700,  1400,  1700,  2100  =  2. 
100,  800,    1500    .     .    =1. 

Examples.  —  March  4, 1877,  was  on  [77  -}-  19  +  4  +0  +  6]^  7, 
remainder  1  =  Sunday.  Jan.  31,  1845  was  on  [45  +  11  +  31  + 
0  +  3]  ^  7,  remainder  6  =  Friday.  Oct.  12, 1492,  was  on  [92  +  23 
+  12  +  2  +  3]  -7-  7,  remainder  6  =  Friday.  Leap  years  are 
known  by  being  divisible  by  4,  except  those  centuries  that  can- 
not be  divided  by  400 ;  hence  1900  was  not  a  leap  year. 

26.  To  find  the  day's  length  at  any  latitude  (for  example, 
71°  N.  Lat.). 

Let  t  be  the  time  before  6  o'clock  for  sunrise ;  then  the  length 
of  the  day  is  (2 1  plus  12)  hours.  If  d  be  the  sun's  declination 
and  I  the  latitude,  then  sin  \  t  equals  cot  (90°  —  T)  tan  d.  For 
longest  day  d  equals  23° 27',  and  I  equals  71°.  Therefore,  sin  \t 
equals  cot  19°  tan  (23° 27').     \t  must  be  expressed  in  degrees. 

log  cot  19°  =  10.463028 

log  tan  (23°  27')=   9.637265 

log  i^  =  10.100293 

As  the  logarithm  of  the  sine  of  an  angle  cannot  be  greater 
than  10,  this  shows  that  the  person's  latitude  is  within  the 
limits  of  the  Arctic  circle,  and  on  the  longest  day  there  the  sun 
does  not  rise  and  set.  —  From  "  The  School  Visitor." 

27.  To  find  the  G.  C.  D.  of  fractions. 

Rule.  —  Find  the  G.  C.  D.  of  the  iiumerators  of  the  fractions, 
and  divide  it  by  the  L.  C.  M.  of  their  denominators. 

28.  To  find  the  L.  C.  M.  of  fractions. 

Rule.  —  Divide  the  L.  C  M.  of  the  numerators  by  the  O.  C  D. 
of  the  denominators. 


MISCELLANEOUS  HELPS  289 

29.  To  find  the  height  of  a  stump  of  a  broken  tree. 

Rule.  —  From  the  square  of  the  height  of  the  tree  subtract  the 
square  of  the  distance  the  top  rests  from  the  base  of  the  tree,  and 
divide  the  remainder  by  twice  the  height  of  the  tree. 

30.  To  find  how  many  board  feet  in  a  round  log. 

Rule.  —  Subtract  4  from  the  diameter  of  the  log  in  inches,  and 
the  square  of  this  remainder  equals  the  number  of  board  feet  in  a 
log  16  feet  long. 

31.  To  find  the  velocity  of  a  nailhead  in  the  rim  of  a  mov- 
ing wheel. 

Rule.  —  Divide  tivice  the  height  of  the  nailhead  above  the  plane 
tipon  which  the  wheel  rolls,  by  the  radius,  and  multiply  this  product 
by  the  velocity  of  the  center;  then  extract  the  square  root. 

Note.  —  Its  velocity  at  the  bottom  is  zero  ;  at  the  top,  twice  that  of  the 
center ;  and  when  its  height  is  half  the  radius,  its  velocity  equals  that  of 
the  center. 

32.  To  find  the  distance  to  the  horizon. 

Rule.  —  Take  one  and  one  half  times  the  height  the  observer  is 
above  the  s^irface  of  the  ground  in  feet.  TJie  square  root  of  this 
number  is  the  number  of  miles  an  object  on  the  surface  can  be  seen, 

33.  Extraction  of  any  root. 

Horner^s  Method,  invented  by  Mr.  Horner,  of  England,  is 
the  best  general  method  of  extracting  roots. 

Any  root  whose  index  contains  only  the  factors  2  or  3  can 
be  extracted  by  means  of  the  square  and  cube  root. 

Rule.  — I.   Divide  the  number  into  periods  of  as  many  figures 

each  as  there  are  units  in  the  index  of  the  root,  and  at  the  left  of 

the  given  number  arrange  the  same  number  of  columns,  ivriting  1 

at  the  head  of  the  left-hand  column  and  ciphers  at  the  head  of  the 

^  others. 

II.  Find  the  required  root  of  the  first  period,  for  the  first  figure 
of  the  root,  multiply  the  number  in  the  1st  col.  by  this  first  term 
of  the  root  and  add  it  to  the  2d  col.,  multiply  this  sum  by  the  root 
and  add  it  to  the  3d  col.,  and  thus  continue,  writing  the  last  prod- 


290 


MATHEMATICAL   WRINKLES 


uct  U7ider  the  first  period;  subtract  and  bring  down   the  next 
period  for  a  dividend. 

III.  Repeat  this  process,  stopping  one  column  sooner  at  the 
right  each  time  until  the  sum  falls  in  the  2d  col.  Then  divide  the 
dividend  by  the  number  in  the  last  column,  which  is  the  trial 
divisor;  the  result  is  the  second  figure  of  the  root. 

IV.  Use  the  second  figure  of  the  root  precisely  as  the  first, 
remembering  to  place  the  products  one  place  to  the  right  in  the 
2d  col.,  two  in  the  3d  col.,  etc.;  continue  this  operation  until  the 
root  is  completed  or  carried  as  far  as  desired. 

Notes.  —  1.  Only  a  part  of  the  dividend  is  used  for  finding  a  root 
figure,  according  to  the  principle  of  place  value.  The  partial  dividend 
thus  used  always  terminates  w^ith  the  first  figure  of  the  period  annexed. 

2.  If  any  dividend  does  not  contain  the  trial  divisor,  place  a  cipher  in 
the  root,  and  bring  down  the  next  period  ;  annex  one  cipher  to  the  last 
term  of  the  2d  column,  two  ciphers  at  the  last  term  of  the  3d,  three  to  the 
4th,  and  then  proceed  according  to  the  rule. 

Example.  —  Extract  the  fourth  root  of  5636405776. 


OPERATION 

0 
2 

0 

4 

8 

12 

12 

(1) 

0 

8 

24 

32  t.  d. 
21063 

56-3640.5776(274 
16 

2 

4 
2 

i03640 

6 

(1) 

24 

609 

3009 

658 

(2) 

53063  T.  D. 
25669 
78732  t.  d. 
1766944 

371441 

2 
(1)  8 

7 

321995776 

87 

3667 
707 

80498944  x.  d. 

321995776 

7 

94 

7 

(2) 

4374 
4336 

101 

7 
(2)  108 

441736 

4 

1084 

—  From  Brooks'  "Higher  Arithmetic." 


MISCELLANEOUS   HELPS  291 


SCIENTIFIC  TRUTHS 


1.  The  intensity  of  light  varies  inversely  as  the  square  of 
the  distance  from  the  source  of  illumination. 

2.  The  intensity  of  sound  varies  inversely  as  the  square  of 
the  distance  from  the  source  of  the  sound. 

3.  Gravitation  varies  inversely  as  the  square  of  the  distance 
between  the  centers  of  gravity. 

4.  The  heating  effect  of  a  small  radiant  mass  upon  a  dis- 
tant object  varies  inversely  as  the  square  of  the  distance. 

MATHEMATICAL  DEFINITIONS 

Algebra  is  that  branch  of  mathematics  in  which  mathemat- 
ical investigations  and  computations  are  made  by  means  of 
letters  and  other  symbols. 

Analytical  Geometry  is  that  branch  of  geometry  in  which  the 
properties  and  relations  of  geometrical  magnitudes  are  investi- 
gated by  the  aid  of  algebraic  analysis. 

Analytical  Trigonometry  is  that  branch  of  trigonometry  which 
treats  of  the  properties  and  relations  of  the  trigonometrical 
functions. 

Applied,  or  Mixed,  Mathematics  is  the  application  of  pure 
mathematics  to  the  mechanic  arts. 

Arithmetic  is  the  science  that  treats  of  numbers,  the  methods 
of  computing  by  them,  and  their  applications  to  business  and 
science. 

Astronomy  is  that  branch  of  applied  mathematics  in  which 
mathematical  principles  are  used  to  explain  astronomical 
facts. 

Calculus  is  that  branch  of  algebraic  analysis  which  com- 
mands, by  one  general  method,  the  most  difficult  problems  of 
geometry  and  physics. 

Calculus  of  Variations  is  that  branch  of  calculus  in  which  the 


292  MATHEMATICAL  WRINKLES 

laws  of  dependence  which  bind  the  variable  quantities  together 
are  themselves  subject  to  change. 

Conic  Sections  is  that  branch  of  Platonic  geometry  which 
treats  of  the  curved  lines  formed  by  the  intersection  of  the 
surface  of  a  right  cone  and  a  plane. 

Descriptive  Geometry  is  that  branch  of  geometry  which  treats 
of  the  graphic  solutions  of  all  problems  involving  three  dimen- 
sions by  means  of  projections  upon  auxiliary  planes. 

Differential  Calculus  is  that  branch  of  calculus  which  investi- 
gates mathematical  questions  by  using  the  ratio  of  certain 
indefinitely  small  quantities  called  differentials. 

Geometry  is  the  science  which  treats  of  the  properties  and 
relations  of  space. 

Gunnery  is  that  branch  of  applied  mathematics  which  treats 
of  the  theory  of  projectiles. 

Integral  Calculus  is  that  branch  of  calculus  which  determines 
the  relations  of  magnitudes  from  the  known  differentials  of 
these  magnitudes.  It  is  the  reverse  method  of  the  differential 
calculus. 

Mathematics  is  that  science  which  treats  of  the  measurement 
of  and  exact  relations  existing  between  quantities  and  of  the 
methods  by  which  it  draws  necessary  conclusions  from  given 
premises. 

Mechanics  is  that  branch  of  applied  mathematics  which  treats 
of  the  action  of  forces  on  material  bodies. 

Mensuration  is  that  branch  of  applied  mathematics  which 
treats  of  the  measurement  of  geometrical  magnitudes. 

Metrical  Geometry  is  that  branch  of  geometry  which  treats 
of  the  length  of  lines  and  the  magnitudes  of  angles,  areas,  and 
solids. 

Navigation  is  that  branch  of  applied  mathematics  which  treats 
of  the  art  of  conducting  ships  or  vessels  from  one  place  to  an- 
other. 


MISCELLANEOUS   HELPS  293 

Optics  is  that  branch  of  applied  mathematics  which  treats  of 
the  laws  of  light. 

Plane  Geometry  is  that  branch  of  pure  geometry  which  treats 
of  figures  that  lie  in  the  same  plane. 

Plane  Trigonometry  is  that  branch  of  trigonometry  which 
treats  of  the  solution  of  plane  triangles. 

Platonic  Geometry  is  that  branch  of  metrical  geometry  in 
which  the  argument,  or  proof,  is  carried  forward  by  a  direct  in- 
spection of  the  figures  themselves,  or  pictured  before  the  eye  in 
drawings,  or  held  in  the  imagination. 

Pure  Geometry  is  that  branch  of  Platonic  geometry  in  which 
the  argument,  or  proof,  uses  compasses  and  ruler  only. 

Pure  Mathematics  treats  of  the  properties  and  relations  of 
quantity  without  relation  to  material  bodies. 

Quaternions  is  that  branch  of  algebra  which  treats  of  the 
relations  of  magnitude  and  position  of  lines  or  bodies  in  space 
by  means  of  the  quotient  of  two  vectors,  or  of  two  directed  right 
lines  in  space,  considered  as  depending  on  four  geometrical 
elements,  and  as  expressible  by  an  algebraic  symbol  of  quadri- 
nomial  form. 

Solid  Geometry,  or  Geometry  of  Space,  is  that  branch  of  pure 
geometry  which  treats  of  figures  which  do  not  lie  wholly 
within  the  same  plane. 

Spherical  Trigonometry  is  that  branch  of  trigonometry  which 
treats  of  the  solution  of  spherical  triangles. 

Surveying  is  that  branch  of  applied  mathematics  which  teaches 
the  art  of  determining  and  representing  areas,  lengths  and  direc- 
tions of  bounding  lines,  and  the  relative  position  of  points  upon 
the  earth's  surface. 

Trigonometry  is  that  branch  of  Platonic  geometry  which  treats 
of  the  relations  of  the  angles  and  sides  of  triangles. 


294  MATHEMATICAL   WRINKLES 

HISTORICAL   NOTES 

The  oldest  known  mathematical  work,  a  papyrus  manuscript 
deciphered  in  1877,  and  preserved  in  the  British  Museum,  was 
written  by  Ash-mesu  (the  moon-born),  commonly  called  Ahmes, 
an  Egyptian,  sometime  before  1700  b.c.  This  work  was  entitled 
"  Directions  for  obtaining  the  Knowledge  of  All  Dark  Things." 
This  work  contains  problems  in  arithmetic  and  geometry  and 
contains  the  first  suggestions  of  algebraic  notation  and  the 
solution  of  equations.  This  work  was  founded  on  another 
work  believed  to  date  back  as  far  as  3400  b.c. 

Pythagoras,  who  died  about  580  e.g.,  raised  mathematics  to 
the  rank  of  a  science.  He  was  one  of  the  most  remarkable 
men  of  antiquity. 

The  study  of  geometry  was  introduced  into  Greece  about 
600  B.C.  by  Thales.  Thales  founded  a  school  of  mathematics 
and  philosophy  at  Miletus,  known  as  the  Ionic  School, 

Euclid's  "  Elements,"  tlie  greatest  textbook  on  geometry, 
was  published  about  300  b.c.  Euclid  taught  mathematics  in 
the  great  university  at  Alexandria,  Egypt. 

The  name  Mathematics  is  said  to  have  first  been  used  by  the 
Pythagoreans. 

About  440  B.C.  Hippocrates  of  Chios  wrote  the  first  Greek 
textbook  on  geometry. 

To  the  great  philosophic  school  of  Plato,  which  flourished  at 
Athens  (429-348  b.c),  is  due  the  first  systematic  attempt  to 
create  exact  definitions,  axioms,  and  postulates,  and  to  distin- 
guish between  elementary  and  higher  geometry. 

Diophantus,  who  died  about  330  a.d.,  was  the  first  writer  on 
algebra  worthy  of  recognition.  His  "Arithmetical  is  the 
earliest  treatise  on  algebra  now  extant.  He  was  the  first  to 
state  that  "  a  negative  number  multiplied  by  a  negative  number 
gives  a  positive  number." 


MISCELLANEOUS   HELPS  295 

Al  Hovarezmi,  who  died  about  830,  published  the  first  book 
known  to  contain  the  word  "  algebra  "  in  the  title. 

The  first  edition  of  Euclid  was  printed  in  Latin  in  1482,  and 
the  first  one  in  English  appeared  in  1570. 

Robert  Recorde  published  the  first  arithmetic  printed  in  the 
English  language  in  1540. 

The  first  arithmetic  published  in  America  was  written  by 
Isaac  Greenwood  and  issued  in  1729. 

Chauncey  Lee  published  in  1797  an  arithmetic  called  "  The 
American  Accomptant."  This  work  contains  the  dollar  mark, 
though  in  much  ruder  form  than  the  character  now  in  use. 

Descartes,  the  French  philosopher,  invented  the  method  of 
computing  graphs  from  equations  about  1637.  On  June  8, 
1637,  he  published  the  first  analytical  geometry. 

The  differential  calculus  was  invented  by  Newton  and 
Leibniz  about  1670. 

In  1686  Leibniz  published  in  a  paper,  "  The  Acta  Erudi- 
torum,"  the  rudiments  of  the  integral  calculus. 

Hipparchus,  who  lived  sometime  between  200  and  100  b.c, 
was  the  greatest  astronomer  of  antiquity  and  originated  the 
science  of  trigonometry. 

The  symbols  of  the  Hindu  or  Arabic  notation,  except  the 
zero,  originated  in  India  before  the  beginning  of  the  Christian 
era.    The  zero  appeared  about  500  a.d. 

Nearly  4000  years  ago  Ahmes  solved  problems  involving  the 
area  of  the  circle  and  found  results  that  gave  ir  =  3.1604.  The 
Babylonians  and  Jews  used  tt  =  3.  The  Romans  used  3  and 
sometimes  4,  or  for  more  accurate  work  S^.  About  500  a.d. 
the_Hindus  used  3.1416.  The  Arabs  about  830  a.d.  used  Vj 
VlO,  3.1416.  In  1596  Van  Ceulen  computed  v  to  over  30  deci- 
mal places.     In  1873  Shanks  computed  v  to  707  decimal  places. 

Logarithms  were  invented  by  John  Napier,  of  Scotland, 
about  1614  A.D.    His  logarithms  were  not  of  ordinary  numbers, 


296  MATHEMATICAL   WRINKLES 

but  of  the  ratios  of  the  legs  of  a  right-angled  triangle  to  the 
hypotenuse. 

Later  Briggs  constructed  tables  of  logarithmic  numbers  to 
the  base  10. 

The  first  publication  of  Briggian  logarithms  of  trigonometric 
functions  was  made  in  1620  by  Gunter.  Gunter  was  a  colleague 
of  Briggs.  He  invented  the  words  cosine  and  cotangent,  and 
found  the  logarithmic  sines  and  tangents  for  every  minute  to 
seven  places. 

HISTORICAL  NOTES  ON  ARITHMETIC 

"  The  Science  of  Arithmetic  is  one  of  the  purest  products  of 
human  thought.  Based  upon  an  idea  among  the  earliest  which 
spring  up  in  the, human  mind,  and  so  intimately  associated 
with  its  commonest  experience,  it  became  interwoven  with 
man's  simplest  thought  and  speech,  and  was  gradually  un- 
folded with  the  development  of  the  race.  The  exactness  of  its 
ideas,  and  the  simplicity  and  beauty  of  its  relations,  attracted 
the  attention  of  reflective  minds,  and  made  it  a  familiar  topic 
of  thought ;  and,  receiving  contributions  from  age  to  age,  it 
continued  to  develop  until  it  at  last  attained  to  the  dignity  of 
a  science,  eminent  for  the  refinement  of  its  principles  and  the 
certitude  of  its  deductions. 

"  The  science  was  aided  in  its  growth  by  the  rarest  minds  of 
antiquity,  and  enriched  by  the  thought  of  the  profoundest 
thinkers.  Over  it  Pythagoras  mused  with  the  deepest  enthu- 
siasm; to  it  Plato  gave  the  aid  of  his  refined  speculations; 
and  in  unfolding  some  of  its  mystic  truths,  Aristotle  employed 
his  peerless  genius.  In  its  processes  and  principles  shines  the 
thought  of  ancient  and  modern  mind  —  the  subtle  mind  of  the 
Hindu,  the  classic  mind  of  the  Greek,  the  practical  spirit  of 
the  Italian  and  English.  It  comes  down  to  us  adorned  with 
the  offerings  of  a  thousand  intellects,  and  sparkling  with  the 


MISCELLANEOUS   HELPS  297 

gems  of  thought  received  from  the  profoundest  minds  of  nearly 
every  age."  —  From  Brooks'  "Philosophy  of  Arithmetic." 

The  first  step  in  the  historical  development  of  arithmetic 
was  in  counting  things.  How  far  back  this  operation  dates  is 
not  known.  Counting  among  primitive  people  was  of  a  very 
elementary  nature,  as  it  is  now  among  people  of  a  low  grade 
of  civilization.  A  knowledge  of  arithmetic  is  coeval  with  the 
race.  Every  people,  no  matter  how  uncivilized,  has  some  crude 
knowledge  of  numbers  and  employs  them  in  its  transactions 
with  one  another.  Some  of  them  have  no  real  numeral 
words,  while  others  have  very  few.  The  Chiquitos  of  Bolivia 
have  no  real  numerals.  The  Campas  of  Peru  have  only 
three,  but  can  count  to  ten.  The  Bushmen  of  South  Africa 
have  but  two  numerals.  The  natives  of  Lower  California  can- 
not count  above  five.  Very  few  of  the  Esquimos  can  count 
above  five.     The  more  intelligent  can  count  to  twenty  or  more. 

The  Egyptians  stand  at  the  beginning  of  the  first  period  in 
the  historical  development  of  arithmetic.  Menes,  their  first 
king,  changed  the  course  of  the  Nile,  made  a  great  reservoir, 
and  built  the  temple  of  Phthah  at  Memphis.  They  built  the 
pyramids  at  a  very  early  period.  Surely  a  people  who  were  en- 
gaged in  enterprises  of  such  magnitude  must  have  known  some- 
thing of  mathematics  —  at  least  of  practical  arithmetic.  To 
them  all  Greek  writers  are  unanimous  in  ascribing,  without 
envy,  the  priority  of  invention  in  the  mathematical  sciences. 

Aristotle  says  that  mathematics  had  its  birth  in  Egypt,  be- 
cause there  the  priestly  class  had  the  leisure  needful  for  the 
study  of  it.  In  Herodotus  we  find  this  (lie  109):  "They 
said  also  that  this  king  (Sesostris)  divided  the  land  among  all 
Egyptians  so  as  to  give  each  one  a  quadrangle  of  equal  size  and 
to  draw  from  each  his  revenues,  by  imposing  a  tax  to  be  levied 
yearly.  But  every  one  from  whose  part  the  river  tore  away 
anything,  had  to  go  to  him  and  notify  what  had  happened ;  he 
then  sent  the  overseers,  who  had  to  measure  out  by  how  much 


298  MATHEMATICAL   WRINKLES 

the  land  had  become  smaller,  in  order  that  the  owner  might 
pay  on  what  was  left,  in  proportion  to  the  entire  tax  imposed. 
In  this  way,  it  appears  to  me,  geometry  originated." 

One  of  the  oldest  known  works  on  mathematics,  a  manuscript 
copied  on  papyrus,  a  kind  of  paper  used  about  the  Mediter- 
ranean in  early  times,  is  still  preserved  and  is  now  in  the  Brit- 
ish Museum.  It  was  deciphered  in  1877  and  found  to  be  a 
mathematical  manual  containing  problems  in  arithmetic  and 
geometry.  It  was  written  by  Ahraes  sometime  before  1700  b.c, 
and  was  founded  on  an  older  work  believed  to  date  back  as  far 
as  3400  B.C.  This  work  is  entitled  "  Directions  for  obtaining 
the  Knowledge  of  All  Dark  Things."  In  the  arithmetical  part 
it  teaches  operations  with  whole  numbers  and  fractions.  Some 
problems  in  this  papyrus  seem  to  imply  a  rudimentary  knowl- 
edge of  proportion.  The  area  of  an  isosceles  triangle,  of  which 
the  sides  measure  10  ruths  and  the  base  4  ruths,  is  erroneously 
given  as  20  square  ruths,  or  half  the  product  of  the  base  by  one 
side.  The  area  of  a  circle  is  found  by  deducting  from  the 
diameter  -i  of  its  length  and  squaring  the  remainder,  tt  is 
taken  =  Q^^-f  =  3.1604. 

According  to  Herodotus  the  ancient  Egyptian  computation 
consisted  in  operating  with  pebbles  on  a  reckoning  board 
whose  lines  were  at  right  angles  to  the  user.  There  is  reason 
to  believe  the  Babylonians  used  a  similar  device.  The  earli- 
est Greeks,  like  the  Egyptians  and  Eastern  nations,  counted 
on  the  fingers  or  with  pebbles.  The  E-omans  employed  three 
methods,  reckoning  upon  the  fingers,  upon  the  abacus  (a  me- 
chanical contrivance  with  columns  for  counters),  and  by  tables 
prepared  for  the  purpose.  The  method  of  finger  reckoning 
seems  to  have  prevailed  among  savage  tribes  from  the  be- 
ginning of  time,  and  every  observer  knows  how  exceedingly 
common  its  use  is  among  children  learning  to  count.  They 
perhaps  adopt  this  method  instinctively. 


MISCELLANEOUS   HELPS  299 

The  Egyptiajis  used  the  decimal  scale.  The  Greeks  and 
Egyptians  made  exclusive  use  of  unit  fractions,  or  fractions 
having  one  for  the  numerator.  They  kept  the  numerator  con- 
stant and  dealt  with  variable  denominators.  The  Babylonians 
kept  the  denominators  constant  and  equal  to  60.  Also  the 
Romans  kept  them  constant,  but  equal  to  12. 

The  Greeks  also  had  much  to  do  with  the  advancement  of 
mathematics.  They  discriminated  between  the  science  of 
numbers  and  the  art  of  calculation.  They  were  among  the 
first  writers  on  arithmetic.  About  twenty-five  centuries  ago 
Pythagoras  classified  numbers  into  perfect  and  imperfect, 
even  and  odd,  solid,  square,  cubical,  etc.  "  He  regarded  num- 
bers as  of  divine  origin  —  the  fountain  of  existence  —  the 
model  and  archetype  of  things  —  the  essence  of  the  universe." 
He  regarded  even  numbers  as  feminine,  and  allied  to  the 
earth ;  odd  numbers  were  supposed  to  be  endued  with  mascu- 
line virtues,  and  partook  of  the  celestial  nature.  He  consid- 
ered "number  as  the  ruler  of  forms  and  ideas,  and  the  cause 
of  gods  and  daemons  " ;  and  again  that  "  to  the  most  ancient 
and  all-powerful  creating  Deity,  number  was  the  canon,  the 
efficient  reason,  the  intellect  also,  and  the  most  undeviating  of 
the  composition  and  generation  of  all  things." 

Philolaus  declared  "that  number  was  the  governing  and 
self-begotten  bond  of  the  eternal  permanency  of  mundane 
natures."  Another  ancient  said  that  number  was  the  judicial 
instrument  of  the  Maker  of  the  universe,  and  the  first  para- 
digm of  mundane  fabrication. 

Plato  ascribed  the  invention  of  numbers  to  God  himself.  In 
the  "  Phaedrus  "  he  said,  "  The  name  of  the  Deity  himself  was 
Theuth.  He  was  the  first  to  invent  numbers,  and  arithmetic, 
and  geometry,  and  astronomy."  In  the  "  Timaeus,"  he  said, 
"Hence,  God  ventured  to  form  a  certain  movable  image  of 
eternity;   and  thus  while  he  was  disposing  the  parts  of  the 


300  MATHEMATICAL   WEINKLES 

universe,  he,  out  of  that  eternity  which  rests  in  unity,  formed 
an  eternal  image  on  the  principle  of  numbers,  and  to  this  we 
give  the  appellation  of  time." 

Euclid,  who  lived  about  300  b.c,  was  one  of  the  early  Greek 
writers  upon  arithmetic.  In  his  "  Elements  "  he  treats  of  the 
theory  of  numbers,  including  prime  and  composite  numbers, 
greatest  common  divisor,  least  common  multiple,  continued 
proportion,  geometrical  progressions,  etc. 

Archimedes,  who  was  born  about  287  e.g.,  was  one  of  the 
most  noted  Greek  mathematicians.  He  discovered  the  ratio  of 
the  cylinder  to  the  inscribed  sphere,  and  in  commemoration  of 
this  the  figure  of  a  cylinder  was  engraved  upon  his  tomb.  He 
also  wrote  two  papers  on  arithmetic.  In  the  first  he  explained 
a  convenient  system  of  representing  large  numbers.  In  the 
second  he  showed  that  this  method  enabled  a  person  to  write 
any  number  however  large,  and  as  proof  gave  his  celebrated 
illustration  that  the  number  of  grains  of  sand  required  to  fill 
the  universe  is  less  than  10^. 

In  1202  Leonardo  of  Pisa  published  his  great  work  "  Liber 
Abaci."  This  work  contained  about  all  the  knowledge  the 
Arabs  possessed  in  arithmetic  and  algebra  and  furnished  the 
most  lasting  material  for  the  extension  of  Hindu  methods. 

In  1540  Kobert  Kecorde  published  the  first  arithmetic  printed 
in  the  English  language.  He  invented  the  present  method  of 
extracting  the  square  root. 

In  1729  Isaac  Greenwood  published  the  first  arithmetic  pub- 
lished in  America. 

In  1788  Nicolas  Pike's  arithmetic  was  published  at  New- 
buryport,  Mass.  It  was  a  very  popular  book  and  was  highly 
recommended  by  George  Washington.  / 

In  1797  Chauncey  Lee  published  "  The  American  Accomp- 
tant." 


MISCELLANEOUS   HELPS  301 

In  1799  Daboll  published  at  New  London,  Conn., "  The  School- 
master's Assistant,"  which  was  indorsed  by  Noah  Webster.  In 
this  book  the  comma  is  used  in  place  of  the  decimal  point. 

In  1821  Warren  Colburn's  "  First  Lessons  in  Intellectual 
Arithmetic "  appeared.  This  book  met  with  remarkable  suc- 
cess. About  two  million  copies  were  sold  in  twenty -five  years. 
It  revolutionized  the  teaching  of  arithmetic,  and  its  influence 
is  felt  to  this  day. 

MATHEMATICAL    SIGNS 

The  symbols  -h  and  —  were  used  by  Widmann  in  his  arith- 
metic published  at  Leipzig  in  1489,  =  by  Kobert  Recorde  in 
his  "Whetstone  of  Witte"  published  in  1557,  x  by  William 
Oughtred  in  1631,  the  dot  (•)  as  a  symbol  of  multiplication  by 
Harriot  in  1631,  the  absence  of  a  sign  between  two  letters  to 
indicate  multiplication  by  Stifel  in  1544,  :  as  a  symbol  of  divi- 
sion by  Leibniz,  /  as  a  symbol  of  division  was  used  very  early 
by  the  Hindus  and  Arabs  and  is  supposed  to  be  the  oldest  of 
all  the  mathematical  signs,  -f-  as  a  symbol  of  division  by  Rahn, 
a  Swiss,  in  an  algebra  published  at  Zurich  in  1659,  >  and  < 
by  Harriot  in  1631,  : :  by  Oughtred  in  1631,  V  was  first  used 
in  this  form  by  Rudoltf  in  1525,  oo  and  fractional  exponents 
by  Wallis  and  Newton  in  1658,  dx  and  J  by  Leibniz  on  October 
29,  1675. 

The  symbols  :^,  >,  <,  indicating  "not  equal,"  etc.,  are 
recent.  Parentheses  were  first  used  as  symbols  of  aggregation 
by  Girard  in  1629.  The  decimal  point  came  into  use  in  the 
seventeenth  century ;  it  seems  to  have  appeared  first  in  a  work 
published  by  Pitiscus  in  1612.  Positive  integral  exponents  in 
the  present  form  were  first  used  by  Chuquet  in  1484. 

The  Greek  letter  v  was  first  used  to  represent  the  ratio  of 
the  circumference  to  the  diameter  by  William  Jones  in  his 
"Synopsis  Palmariorum  Matheseos,"  in  1706,  and  came  into 
general  use  through  the  influence  of  Euler. 


302 


MATHEMATICAL   WRINKLES 


BB 


e  g  o 


o  be 


bh 


AhoqO     ^-^ 


a   -43 


Ph      Q    H 


W        .2 
W         2 


5     'S  a 


1l 


c  a>  rt  S  t*,  tc  S? 
-g  S  «  «  ee  C2  " 


■r^  2^  CO  rjt  lO  «0  t- C 


be 


^  OQ 


V.    -2 


Tl        (M        CO 


^    s     1 1: 


MISCELLANEOUS   HELPS 


303 


'4 

Bg«  3  V  =-9 


•3  5  =  S  c 


1    I 
I.    I 

fi    e 


II 


E  IT  fc  I-  — 

a  e  =  4*^ 


e   2   s 

03    a    vj 


I  I 


.5    5 


a 

.  I 


H     H 


oc^tnoh- 


E       ®      ft. 


1  1   «l  ^  . 

Ph     O       O       (S     H 


II 


TABLES 


O 

>0 

iOO»CO»OQiCO»COiOO»00»OOiOO»riOiOO»r     1 

C4 

00 

T-iT-iT-ir-i(Mc^j(MC-<cococ'5r:)TH-tirtiTriio»cob;S 

CO 

ssSSs2liigliSiiS|8||||g 

8sa22ss§iiiiiigis|||g|| 

C4 

1^ 

S3|§3lSi8i||Sim§5|||| 

^ 

§ 

^ 

OOOOOOOQOOOOOOOOOO 

^^888 

Tt^    ■^    Tti    rr    lO 

O 

0) 

^ 

rHrHr-lr-lT-l(MC^J(MC^|lMCOCCCCCOCO 

8  $S  ^g  ^ 

cc    rh    •*    ^    Tt^ 

oo 

r-l 

§§ 

ssssg^igSsiiiiiiii 

X    :c   ■*  (M  o 

^  i  5J  ^  ^ 

00 

i-H 

^. 

3SSS2gggg|gi||S|||| 

t~"     Tfl    rH    CO    m 

tH 

CO 
ft 

M 

r-lr-lT-ii-iT-(T-IC^C4(M<NC^C<lIOCO 

^  S  §  ':^  8 
«  ^  ^  c§  8 

CO 
I-H 

^ 

i^S^S|||||||g  §11111 

15  ^  ^  S  {2 

CO    fO    JS    cc    CO 

rH 

85 

^§g^^gllgllllgllll 

i  i%'  1  § 

rH 

00 

r-l 

^ 

p^^gs^ggigiiiiimi 

glili 

CO 

^ 

J8^S^^^§^??^ 

||g|SS|| 

(M    "*(    CO    00    O 

rH 

rH 
rH 

rl     .    1     .    1     ■    1 

rH 

g]??^13S^§88gg| 

r-lr-tT-HTHr-(i-iC^4C<I 

??  ^  £?  ^  {2 

!M    (N    (N    C>J    CM 

o 

rH 

S 

S§§Sg88|g| 

Siligiii 

§liii 

o 

rH 

o» 

oo 

^  §S  ij  iS  8  ?3  S3  8  8  1 

3§I3§ISI 

8^gS^ 

rH    r-(    (N    (M    iM 

Oi 

00 

CO 

rH 

^^^^§^SS28S 

Igggl^ss 

||g|8 

00 

t* 

T*< 

?^^^^^5§Sg^^ 

g  g  1  g  s  1 1  g 

rh    S    CO    S    t^ 
rH    rH    rH     T-H     r-< 

t- 

CD 

c-i 

S^g§8^^^S8?2 

E2^88§8^g 

^    ^^    ^     ^     g 

co 

I  i    .  ^    1   .    1  ( 

lO 

o 

J3  g  ^^  ^  1-?  ^  ^  g  i§  S 

§gl2SS8g^8 

3S  J28^ 

rH    rH    r-1    rH    rH 

lO 

1 

cc 

A3Sg^§5^§8^^^ 

Sg8^SS£5g 

^28^88 

«# 

"1 

o 

^?3J5^5^^^8?S^ 

^^^^^^^S 

?S  8  8  ^  ^  1  « 

«! 

■* 

^'^^E2:^SSgS^c5 

C^(M8cO^COCOTtl 

^  ^  '^  ^  8  1  « 

rH 

C4 

CO^WatOl'OOCSOrHC* 

eo^>ocot*ooo>0 

rH    «    CO    Tt<    <0 
C4  (M  <M  (M  O) 

rH 

r-t    r-t    T-t 

'"''"''"''"'        "^ 

304 


TABLES 


305 


Table  showing  the  Amouxt  of  SI  at  Compound  Interest  fbom 
1  Year  to  20  Years 


Yb. 

1 

24  Peb  Cent 

8  Peb  Cent 

SJPebCe.^t 

4  Pkb  Cent 

5  Pek  Cent 

6  Peb  Cent 

1.025 

1.03 

1.035 

1.04 

1.06 

1.06 

2 

1.0o0«2o 

1.0609 

1.071225 

1.0816 

1.1026 

1.1236 

8 

1.070891 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

4 

1.103813 

1.125509 

1.147523 

1.169859 

1.215500 

1.202477 

5 

1.131408 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

6 

1.159603 

1.1940.52 

1.229255 

1.265319 

1.340096 

1.418619' 

7 

1.18868(3 

1.229874 

1.272279 

1.315932 

1.4071 

1.50363 

8 

1.218103 

1.20677 

1.316809 

1.3685(59 

1.477455 

1.593848 

9 

1.248863 

1.304773 

1.302897 

1.423312 

1  651328 

1.(589479 

10 

1.280085 

1.343916 

1.410599 

1.480244 

1.628895 

1.790848 

11 

1.312087 

1.384234 

1.45997 

1.5394W 

1.710339 

1.898299 

12 

1.344889 

1.425761 

1.611069 

1.001032 

1.796866 

2.012197 

13 

1.378511 

1.4085:34 

1.563956 

1.665074 

1.885649 

2.132928 

14 

1.412974 

1.51259 

1.018095 

1.731676 

1.979932 

2.260904 

15 

1.448298 

1.567967 

1.075349 

1.800944 

2.078928 

2.396558 

16 

1.484506 

1.604706 

1.733986 

1.872981 

2.182875 

2.640862 

17 

1.521618 

1.652848 

1.794076 

1.947901 

2.292018 

2.692773 

18 

1.559659 

1.702433 

1.857489 

2.025817 

2.406619 

2.854339 

19 

1.59865 

1.753506 

1.922501 

2.106849 

2.52695 

3.0266 

20 

1.638616 

1.806111 

1.989789 

2.191123 

2.653298 

8.207130 

Tb. 

7  Peb  Cknt 

S  Peb  Cent 

9  Peb  Ckxt 

It)  Peb  Cent 

11  Peb  Cent 

12  Peb  Cent 

1 

1.07 

1.08 

1.09 

1.10 

1.11 

1.12 

2 

1.1449 

1.1664  • 

1.1881 

1.21 

1.2321 

1.2544 

8 

1.225043 

1.259712 

1.295029 

1.331 

1.3(37031 

1.404908 

4 

1.310790 

1.360489 

1.411682 

1.4641 

1.51807 

1.573519 

6 

1.402552 

1.469328 

1.638624 

1.61051 

1.685058 

1.702342 

6 

1.50073 

1.686874 

1.6771 

1.771561 

1.870414 

1.973822 

7 

1.60.')781 

1.713824 

1.828039 

1.948717 

2.07616 

2.210681 

8 

1.718186 

1.85093 

1.992503 

2.143589 

2.304537 

2.475963 

9 

1.838459 

1.999005 

2.171893 

2.357948 

2.568036 

2.773078 

10 

1.967161 

2.158926 

2.367364 

2.593742 

2.a3942 

3.106848 

11 

2.104852 

2.331639 

2.680426 

2.853117 

8.151757 

3.478649 

12 

2.252192 

2.61817 

2.812666 

3.1.38428 

3.49845 

3.895975 

18 

2.409845 

2.719624 

3.065806 

3.452271 

3.883279 

4.363492 

14 

2.6785:W 

2.a37194 

3.341727 

3.797498 

4.31044 

4.887111 

15 

2.759031 

3.172169 

3.642482 

4.177248 

4.784688 

6.473565 

16 

2.952164 

3.426943 

3.97030({ 

4.594973 

6.310893 

6.130392 

17 

3.158816 

3.700018 

4.327633 

6.06447 

5.896091 

6.86604 

18 

3.379932 

3.990019 

4.71712 

5.659917 

6.543561 

7.689964 

19 

3.616527 

4.315701 

6.141661 

6.115909 

7.263342 

8.61276 

20 

3.809684 

4.660957 

5.604411 

6.7275 

8.062309 

9.646291 

306 


MATHEMATICAL   WRINKLES 


Scalene  Triangles  whose 

Areas  and  Sides  are 

Integral 


Right  Triangles    whose  Sides 
are  Integral 


4 

13 

15 

20 

37 

51 

13 

14 

15 

25 

39 

66 

7 

15 

20 

25 

52 

63 

11 

13 

20 

25 

61 

52 

.10 

17 

21 

25 

74 

77 

12 

17 

25 

26 

51 

55 

13 

20 

21 

29 

52 

69 

17 

25 

26 

34 

65 

93 

17 

25 

28 

35 

63 

66 

13 

37 

40 

36 

61 

65 

13 

40 

45 

37 

91 

66 

15 

34 

35 

39 

41 

50 

15 

37 

44 

39 

85 

92 

17 

39 

44 

40 

51 

77 

25 

29 

36 

41 

51 

58 

25 

39 

40 

41 

84 

85 

29 

35 

48 

48 

85 

91 

39 

41 

50 

50 

69 

73 

13 

68 

75 

51 

52 

53 

15 

41 

52 

52 

73 

75 

17 

55 

60 

43 

61 

68 

3 

4 

5 

66 

88 

110 

6 

8 

10 

69 

92 

115 

9 

12 

15 

72 

96 

120 

12 

16 

20 

75 

100 

125 

15 

20 

25 

78 

104 

130 

18 

24 

30 

81 

108 

135 

21 

28 

35 

84 

112 

140 

24 

32 

40 

87 

116 

145 

27 

36 

45 

90 

120 

150 

30 

40 

50 

93 

124 

155 

33 

44 

55 

96 

128 

160 

36 

48 

60 

99 

132 

165 

39 

52 

65 

102 

136 

170 

42 

56 

70 

105 

140 

175 

45 

60 

75 

108 

144 

180 

48 

64 

80 

111 

148 

185 

51 

68 

85 

114 

152 

190 

54 

72 

90 

117 

156 

195 

57 

76 

95 

120 

160 

200 

60 

80 

100 

123 

164 

205 

63 

84 

105 

126 

168 

210 

Squares  of  Integers  from  10  to  100 


No. 
10 

0 

1 

2 

3 

4 

6 

6 

7 
289 

8 
324 

9 

100 

121 

144 

169 

196 

225 

256 

361 

20 

400 

441 

484 

529 

576 

625 

676 

729 

784 

841 

30 

900 

961 

1024 

1089 

1156 

1225 

1296 

1369 

1444 

1521 

40 

1600 

1681 

1764 

1849 

1936 

2025 

2116 

2209 

2304 

2401 

50 

2500 

2601 

2704 

2809 

2916 

3025 

3136 

3249 

3364 

3481 

60 

3600 

3721 

3844 

3969 

4096 

4225 

4356 

4489 

4624 

4761 

70 

4900 

5041 

5184 

5329 

5476 

5625 

5776 

5929 

6084 

6241 

80 

6400 

6561 

6724 

6889 

7056 

7225 

7396 

7569 

7744 

7921 

90 

8100 

8281 

8464 

8649 

8836 

9025 

9216 

9409 

9604 

9801 

TABLES 


307 


Square  Roots  of  Numbers  from  0  to  10,  at  Intervals  of  .1 


Ko. 
0 

.0 

.1 

,2 

8 

.4 

.5 

.6 

.776 

.7 

.8 

.0 

0 

.316 

.447 

.648 

.632 

.707 

.837 1    .894 

.949 

1 
2 
3 

1.000 
1.414 
1.732 

1.049 
1.449 
1.761 

lAYX, 
1.789 

1.140 
1.517 
1.817 

1.183 
1.549 
1.844 

1.225 
1.581 
1.871 

1.265 
1.612 
1.897 

1.304 
1.643 
1.924 

1.342 
1.073 
1.149 

1.378 
1.703 
1.975 

4 
5 
6 

2.000 
2.286 
2.449 

2.025 
2.258 
2.470 

2.049 
2.280 
2.490 

2.074 
2.302 
2.510 

2.098 
2.324 
2.530 

2.121 
2.345 
2.550 

2.145 
2.366 
2.569 

2.168 
2.387 
2.588 

2.191 
2.408 
2.608 

2.214 
2.429 
2.627 

7 
8 
9 

2.646 
2.828 
3.000 

2.665 
2.846 
3.017 

2.683 
2.864 
3.033 

2.702 
2.881 
3.050 

2.720 
2.898 
3.066 

2.739 
2.915 
3.082 

2.757 
2.933 
3.098 

2.775 
2.950 
3.114 

2.793 
2.966 
3.130 

2.811 
2.983 
3.146 

Square  Roots  of  Integers  from  10  to  100 


No. 

0 

1 

2 

8 

4 

6 

6 

7 

8 

• 

10 
20 
30 

3.162 
4.472 
5.477 

3.317 
4.583 
6.668 

3.464 
4.690 
6.667 

3.606 
4.796 
6.745 

3.742 
4.899 
6.831 

3.873 
6.000 
6.916 

4.000 
5.099 
6.000 

4.123 
6.196 
6.083 

4.243 
6.292 
6.164 

4.359 
6.385 
6.245 

40 
60 
60 

6.326 
7.071 
7.746 

6.403 
7.141 
7.810 

6.481 
7.211 
7.874 

6.567 
7.280 
7.937 

6.633 
7.348 
8.000 

6.708 
7.416 
8.062 

6.782 
7.483 
8.124 

6.856 
7.550 
8.185 

6.928 
7.616 
8.246 

7.000 
7.681 
8.307 

70 
80 
90 

8.367 
8.944 
9.487 

8.426 
9.000 
9.639 

8.486 
9.056 
9.692 

8.644 
9.110 
9.644 

8.602 
9.165 
9.695 

8.660 
9.220 
9.747 

8.718 
9.274 
9.798 

8.775 
9.327 
9.849 

8.8.S2 
9.381 
9.899 

8.888 
9.434 
9.950 

Cube  Rck)ts  of  Ixtec.krs  from  1  to  30 


No. 

Cube  Root 

No. 

CUBK   EOOT 

No. 

ClfHK    Ko«.T 

1 

1.000000 

11 

2.223980 

21 

2.758024 

2 

1.255)021 

12 

2.289420 

22 

2.802030 

1.442250 

13 

2.351335 

23 

2.843867 

1.587401 

14 

2.410142 

24 

2.884409 

1.70S)076 

15 

2.466212 

26 

2.024018 

1.817121 

16 

2.519842 

26 

2.962496 

1.912931 

17 

2.571282 

27 

3.000000 

2.000000 

18 

2.620741 

28 

3.036589 

9 

2.080084 

19 

2.668402 

29 

3.072317 

10 

2  154435 

20 

2.714418 

30 

3.107233 

308 


MATHEMATICAL   WRINKLES 


Tables  of  Prime  Numbers  from  1  to  1000 


1 

i09 

^69 

^39 

^17 

811 

2 

13 

71 

43 

19 

21 

3 

27 

77 

49 

31 

23 

5 

31 

81 

57 

41 

27 

7 

37 

.  83 

61 

43 

29 

11 

39 

93 

63 

47 

39 

13 

49 

507 

67 

53 

53 

17 

51 

11 

79 

59 

57 

19 

57 

13 

87 

61 

59 

23 

63 

17 

91 

73 

63 

29 

67 

31 

99 

77 

77 

31 

73 

37 

503 

83 

81 

37 

79 

47 

09 

91 

83 

41 

81 

49 

21 

701 

87 

43 

91 

53 

23 

09 

507 

47 

93 

59 

41 

19 

11 

53 

97 

67 

47 

27 

19 

59 

99 

73 

57 

33 

29 

61 

ni 

79 

63 

39 

37 

67 

23 

83 

69 

43 

41 

71 

27 

89 

71 

51 

47 

73 

29 

97 

77 

57 

53 

79 

33 

^01 

87 

61 

67 

83 

39 

09 

93 

69 

71 

89 

41 

19 

99 

73 

77 

97 

51 

21 

601 

87 

83 

iOl 

57 

31 

07 

97 

91 

03 

63 

33 

13 

809 

97 

07 

Note.  —  The  hundreds'  digits  are  not  repeated  after  being  first  intro- 
duced, unless  at  the  heads  of  columns. 


Constants 


=  3.14159265359 
=  0.7853982 

=  0.5235988 


0.3183099 

9.8696044 
0.1013212 


V7r  =  1.7724539 

1 


0.5641896 


Vt 


log 


0.4971499 


log  -  =  9.8950899  -  10 


^6 
log  - 

log  tt'^ 


9.7189986-  10 

9.5028501  -  10 
0.9942997 


log  -L  =  9.0057003  -  10 

log  Vtt  =  0.2485749 
logA:  =  9.7514251 -10 


TABLES 


309 


Specific  Gravities.  —  Water  1 

A  table  showing  the  weight  of  each  substance  compared  with  an  equal 
volume  of  pure  water.  A  cubic  foot  of  rain  water  weighs  1000  ounces, 
or  02i  lb.  Avoir.  To  find  the  weight  of  a  ctibic  foot  of  any  substance 
named  in  the  table,  move  the  decimal  point  three  places  toward  the 
right,  which  is  multiplying  by  1000,  and  the  result  will  show  Qie  number 
of  ounces  in  a  cubic  foot. 


BCB9TAXCKS 


Acid,  acetic      .... 

Acid,  nitric 

Acid,  sulphuric     .     .     . 

Air 

Alcohol,  of  commerce   . 
Alcohol,  pure  .     .     .     . 

Alder  wood 

Ale 

Alum 

Aluminum 

Amber  

Amethyst 

Ammonia 

Ash 

Blood,  human  .... 
Brass  ....  (about) 

Brick 

Butter 

Cedar 

Cherry 

Cider  

Coal,  bituminous  (about) 
Coal,  anthracite  .     .     . 

Copper    

Coral 

Cork    .     . 

Diamond 

Earth  (mean  of  the  globe) 

Elm 

Emerald 

Fir 

Glass,  flint  ...     .     .     . 

Glass,  plate      .... 

Gold,  native     .... 

Gold,  pure,  cast    .     .     . 

Gold,  coin 

Granite 

Gum  Arabic     .... 

Gypsum 

Honey 

Ice 

Iodine 

Iron 

Iron,  ore      

Ivory  

Laid 


Specific  Grav. 


1.008 

1.271 

1.841  to  2.125 

.001227 

.835 

.794 

.800 
1.035 
1.724 
2.560 
1.064 
2.750 

.875 
8.400 
1.054 
8.400 
2.000 

.912 
.457  to  ^661 

.715 
1.018 
1.250 
1.600 
8.788 
2.540 

.240 
3.630 
6.210 

.661 
2.678 

.550 

2.760 

2.7f50 

15.600  to  19..of)0 

lO.'ioH 

17.(>47 

2. «).'>'-' 

2.288 
1.456 

.930 
4.948 
7.645 
4.900 
1.917 

.947 


SfBSTANCES 


Lead,  cast .    . 

Lead,  white    . 

Lead,  ore    .     . 

Lignum  vitse  . 

Lime  .... 

Lime,  stone    . 

Mahogany  .     . 

Manganese 

Maple     .     .     . 

Marble   .     .     . 

Men  (living)    . 

Mercury,  pure 

Mica  .... 

Milk  .... 

Nickel    .     .     . 

Niter.     .     .     . 

Oil,  castor  .     . 

Oil,  linseed     . 

Opal  .... 

Opium    .     .     . 

Pearl.    .    .    . 

Pewter  ... 

Platinum  (native) 

Platinum,  wire 

Poplar    .    .    . 

Porcelain    .    • 

Quartz   .    .     . 

Rosin      .     .     . 

Salt    .... 

Sand  .... 

Silver,  cast 

Silver,  coin 

Slate  .... 

Steel  .... 

Stone      .     .     . 

Sulphur,  fused 

Tallow   .     .     . 

Tar    .... 

Tin     .     .     .     . 

Turpentine,  spirits  of 
i  Vinegar      .     . 
1  Walnut  .     .     . 
;  Water,  distilled 
!  Water,  sea 
!  Wax  .... 

Zinc,  cast   .     . 


Si'ECinc  GBA^ 


ll.;i5() 

7.2;« 

7.250 

1.333 

.804 

2.386 

1.063 

3.700 

.750 

2.716 

.891 

14.000 

2.750 

1.032 

8.279 

1.900 

.970 

.940 

2.114 

l..'i37 

2.510 

7.471 

17.000 

21.041 

.383 

2.385 

2.500 

1.100 

,     2.130 

1.5O0  to  1 .800 

10.474 

10.534 

2.110 

7.816 

2.000  to  2.700 

um 

.941 
1.015 
7.291 

.870 
1.013 

.671 
1.000 
1.028 

.897 
7.190 


310 


MATHEMATICAL   WEINKLES 


Approximate  Values  of  Foreign  Coins  in  United  States  Monet 


Value  in 

Country 

Standard 

Monetary  Unit 

Terms  of 

U.  S.  Gold 

Dollars 

Argentine  Republic   . 

Gold  &  Silver 

Peso 

.965 

Austria- Hungary  .     . 

Gold 

Crown 

.203 

Belgium 

Gold  &  Silver 

Franc 

.193 

Bolivia 

Silver 

Boliviano 

.441 

Brazil 

Gold 

Milreis 

.546 

British  Possessions  in 

N.  A.  [except  New- 

foundland^   .     .     . 

Gold 

Dollar 

1.00 

Central  Am.  States 

Guatemala"] 

Honduras 

Silver 

Peso 

.441 

Nicaragua  ( 

Salvador    J 

Chili 

Gold 

Peso 

.365 

[  Shanghai 

.661 

China 

Silver 

TaeU  Haikwan 

.736 

[  Canton 

.722 

Colombia      .... 

Gold 

Dollar 

1.00 

Costa  Rica   .... 

Gold 

Colon 

.465 

Cuba 

Gold 

Peso 

.91 

Denmark      .... 

Gold 

Crown 

.268 

Ecuador 

Gold 

Sucre 

.487 

Egypt 

Gold 

Pound  [100  Piastres] 

4.943 

Finland 

Gold 

Mark 

.193 

France     

Gold  &  Silver 

Franc 

.193 

German  Empire    .     . 

Gold 

Mark 

.238 

Great  Britain    .     .     . 

Gold 

Pound  Sterling 

4.866^ 

Greece 

Gold  &  Silver 

Drachma 

.193 

Haiti 

Gold  &  Silver 

Gourde 

.965 

India 

Gold 

Pound  Sterling 

4.866^ 

Italy 

Gold  &  Silver 

Lira 

.193 

Japan       

Gold 

Yen,  Gold 

.498 

Liberia 

Gold 

Dollar 

1.00 

Mexico 

Gold 

Peso 

.498 

Netherlands      .     .     . 

Gold  &  Silver 

Florin 

.402 

Newfoundland  .     .     . 

Gold 

Dollar 

1.014 

Norway 

Gold 

Crown 

.268 

Peru 

Gold 

Sol 

.487 

Portugal 

Gold 

Milreis 

1.08 

Russia 

Gold 

Rouble,  Gold 

.515 

Spain 

Gold  &  Silver 

Peseta 

.193 

Sweden 

Gold 

Crown 

.268 

Switzerland       .     .     . 

Gold  &  Silver 

Franc 

.193 

Tripoli 

Silver 

Mahbub  [20  Piastres] 

.413 

Turkey 

Gold 

Piastre 

.044 

Venezuela    .... 

Gold  &  Silver 

Bolivar 

.193 

TABLES  311 

WEIGHTS  AND  MEASURES 
Avoirdupois  Weight 

16  ounces  (oz.)  =  1  pound  (lb.) 

100  pounds  =  1  hundredweight  (cwt.) 

20  hundredweight,  or  2000  pounds  =  1  ton  (T.) 


1  ton  =  20  cwt.  =  2000  lb.  =  32,000  oz. 
1  pound  Avoirdupois  weight  =  7000  grains. 
1  ounce  Avoirdupois  weight  =  437  J  gr. 

Troy  Weight 

24  grains  (gr.)      =  1  pennyweight  (pwt.) 
20  pennyweights  =  1  ounce  (oz.) 
12  ounces  =  1  pound  (lb.) 


1  lb.  =  12  oz.  =  240  pwt.  =  5760  gr. 
1  ounce  Troy  weight  =  480  gr. 


Apothecaries*  Weight 

20  grains  (gr.  xx)  =  1  scruple  (sc.,  or  3) 
3  scruples  (3iij)  =  1  dram  (dr.,  or  3) 
8  drains  (3viij)   =  1  ounce  (oz.,  or  3) 

12  ounces  (3xij)   =1  pound  (lb.,  or  ft.) 


1  ft.  =  12  3  =  96  3  =  288  3  =  5760  gr. 
Medicines  are  bought  and  sold  in  quantities  by  Avoirdupois  weight. 

Apothecaries'  Fluid  Measure 

60  minims,  or.  drops  (HI,  or  gtt.)  =  1  fluidrachm  (f  3) 
8  fluidrachms  =  1  fluidounce  (f  3  ) 

16  fluidounces  =  1  pint  (O.) 

8  pints  =  1  gallon  (Cong.) 


1  Cong.  =  80.  =  128  f  3  =  1024  f  3  =  61,440  m. 
O.  is  an  abbreviation  of  octans,  the  Latin  for  one  eighth  ;  Cong,  for  con- 
giarium,  the  Latin  for  gallon. 


312  MATHEMATICAL   WRINKLES 

Linear  Measure 

12    inches  (in.)  =  1  foot  (ft.) 

3    feet  =  1  yard  (yd.) 

5^  yards,  or  16|  feet  =  1  rod  (rd.) 

320    rods  =  1  mile  (mi.) 


1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft.  =  63,300  in. 

Mariners'   Linear  Measure 

9    inches  (in.)        =1  span  (sp.) 
8    spans,  or  6  feet  =  1  fathom  (fath.) 
120    fathoms  =1  cable's  length  (c.  1.) 

7^  cable  lengths      =  1  nautical  mile  (or  knot)  (mi.) 
3   miles  =  1  league 

Geographical  and  Astronomical  Linear  Measure 

1       geographic  mile  =  1.15  statute  miles 

3       geographic  miles         =  1  league 
60       geographic  miles,  or  1  _  f  of  latitude  on  a  meridian, 

69.16  statute  miles  J  ~  [or  of  longitude  on  the  equator 

360       degrees  =  the  circumference  of  the  earth 

Surveyor's  Linear  Measure 

7.92  inches  =  1  link  (1.) 
25  links    =  1  rod  (rd. )  \ 

4  rods     =  1  chain  (ch.) 
80  chains  =  1  mile  (mi.) 


1  mile  =  80  ch.  =  320  rd.  =  8000  1.  =  63,360  in. 

Jewish  Linear  Measure 

cubit  =  1.824  ft.      I      mile  (4000  cubits)  =  7296  ft. 

Sabbath  day's  journey  =  3648  ft.       |      day's  journey  =  33.164  mi. 

Square  Measure 

144    square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9    square  feet  =  1  square  yard  (sq.  yd.) 

SO^  square  yards  =  1  square  rod  or  perch  (sq.  rd.;  P.) 

160    square  rods  =  1  acre  (A.) 

640    acres  =  1  square  mile  (sq.  mi. ) 


TABLES  313 

•q.  mL   A.      sq.  rd.  sq.  yd.          sq.  ft.  sq.  in. 

1  =  640  =  102,400  =  3,097,600  =  27,878,400  =  4,014,489,600 

1  =    160  =  4840  =    43,660  =  6,272,640 

1  =  30J  =      272J  =  89,204 

1  =        9   =        1290 

Surveyor's  Square  Measure 
626  square  links  (sq.  1.)  =  1  square  rod  (sq.  rd.) 

16  square  rods  =  1  square  chain  (sq.  ch.) 

10  square  chains  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

36  square  miles  =  1  township  (Tp.) 

Tp.    sq.  mi.           A.  sq.  ch.  sq.  rd.  sq.  1. 

1  =  36  =  23,040  =  230,400  =  3,686,400  =  2,304,000,000 

1  =        640  =  6400  =      102,400  =  6,400,000 

1  =  10  =             160  =  100,000 

Cubic  Measure 
1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

1  cu.  yd.  =  27  cu.  ft.  =  46,666  cu.  in. 

Wood  Measure 
16  cubic  feet         =  1  cord  foot  (cd.  ft.) 

8  cord  feet,  or  ]       ,        ,  ... 

128  cubic  feet      1  =  1'=""^  ^'^'^ 

24J  cubic  feet        =  I  PercMPch.)  of  etone 
*  [  or  of  masonry 

Drt  Measure 

2  pints  (pt.)  =  1  quart  (qt.) 

8  quarts  =  1  peck  (pk.) 
4  pecks   =  1  bushel  (bu.) 
1  bu.  =  4  pk.  =  32  qt.  =  64  pt. 

Liquid  Measure 
4    gills      =  1  pint  (pt.) 
2    pints     =  1  quart  (qt.) 
4   quarts  =  1  gallon  (gal.) 
31i  gallons  =  1  barrel  (bbl.) 
1  bbl.  =  31 J  gal.  =  126  qt.  =  262  pt.  =  1008  gi. 


3l4  MATHEMATICAL   WRINKLES 

Circular  Measure 
60  seconds  =  1  minute  (') 
60  minutes  =  1  degree  (°) 
360  degrees  =  1  circle 

Commercial  Weight 
16  drams    =  1  ounce  (oz.) 
16  ounces    =  1  pound  (lb.) 
2000  pounds  =  1  ton  (T.) 

Paper 
24  sheets     =  1  quire 
20  quires     =  1  ream 
2  reams     =  1  bundle 
6  bundles  =  1  bale 

English  Money 
4  farthings  (far.)  =  1  penny  (d.) 
12  pence  =  1  shilling  (s.) 

20  shillings  =  1  pound  (£) 


1  £  =  20s.  =  240d.  =  960  far. 
1  £  =  $4.8665  in  U.  S.  money 

Measure  of  Time 
60  seconds  (sec. )  =  1  minute  (min. ) 
60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da. ) 

7  days  =  1  week  (wk.) 

365  days  =  1  year  (yr.) 

366  days  =  1  leap  year 

1  da.  =  24  hr.  =  1440  min.  =  86,400  sec. 

THE  METRIC  SYSTEM 

(^The  Acme  of  Simplicity} 
The  following  prefixes  are  used  in  the  Metric  System  : 

(Greek)  (Latin) 

deka,  meaning         10  deci,  meaning  .1 

hekto,  meaning      100  centi,  meaning  .01 

kilo,  meaning       1000  milli,  meaning  .001 
myria,  meaning  10,000 


TABLES  315 


Linear  Measure 


10  millimeters  (mm.)  =  1  centimeter  (cm.) 

10  centimeters  =  1  decimeter  (dm.) 

10  decimeters  =  1  meter  (m. ) 

10  meters  =  1  dekameter  (Dm.) 

10  dekametere  =  1  hektometer  (Hm.) 

10  hektometers  =  1  kilometer  (Km.) 

10  kilometers  =  1  myriameter  (Mm.) 

SgcAUK  Measure 

100  square  millimeters  (sq.  mm.)  =  1  square  centimeter  (sq.  cm.) 
100  square  centimeters  =  1  square  decimeter  (sq.  dm.) 

100  square  decimeters  =  1  square  meter  (sq.  m.) 

100  square  meters  =  1  square  dekameter  (sq.  Dm.) 

100  square  dekameters  =  1  square  hektometer  (sq.  Hm.) 

100  square  hektometers  =  1  square  kilometer  (sq.  Km.) 

The  area  of  a  farm  is  expressed  in  hektares. 

The  area  of  a  country  is  expressed  in  square  kilometers. 

Cubic  Measure 

1000  cubic  millimeters  (cu.  mm.)  =  1  cubic  centimeter  (cu.  cm.) 
1000.  cubic  centimeters  =  1  cubic  decimeter  (cu.  dm.) 

1000  cubic  decimeters  =  1  cubic  meter  (cu.  m.) 

Table  of  Capacity 

10  milliliters  (ml.)  =  1  centiliter  (cl.) 

10  centiliters  =  1  deciliter  (dl.) 

10  deciliters  =  1  liter  (1.) 

10  liters  =  1  dekaliter  (Dl.) 

10  dekaliters  =  1  hektoliter  (HI.) 

.10  hektoliters  =  1  kiloliter  (Kl.) 

10  kiloliters  =  1  myrialiter  (Ml.) 


The  hektoliter  is  used  in  measuring  grain,  vegetables,  etc. 
The  liter  is  used  in  measuring  licjuids  and  small  fruits. 

Table  of  Weight 

10  milligrams  (mg.)  =  1  centigram  (eg.) 
10  centigrams  =  1  decigram  (dg.) 

10  decigrams  =  1  gram  (g.) 


316  MATHEMATICAL  WRINKLES 

10  grams  =  1  dekagram  (Dg. ) 

10  dekagrams  =  1  hektogram  (Hg.) 

10  hektograms  =  1  kilogram  (Kg.) 

1000  kilograms  =  1  metric  ton  (T.) 


A  myriagram    =  10,000  grams 
A  quintal  (Q.)  =  100,000  grams 

TABLE  OF  EQUIVALENTS 

Long  Measure 

1  inch  =  2.54  centimeters  1  centimeter  =    .3937  of  an  inch 

1  foot  =    .3048  of  a  meter  1  decimeter   =    .328  of  a  foot 

1  yard  =    .9144  of  a  meter  1  meter  =  1.0936  yards 

1  rod    =  6.029  meters  1  dekameter  =  1.9884  rods 

1  mile  =  1.6093  kilometers  1  kilometer    =  .62137  of  a  mile 

Square  Measure 

1  square  inch  =    6.462  square  centimeters 
1  square  foot  =      .0929  of  a  square  meter 
1  square  yard  =      .8361  of  a  square  meter 
1  square  rod    =  26.293  square  meters 
1  acre  =  40.47  ares 

1  square  mile  =  259  hectares 

1  square  centimeter  =    .155  of  a  square  inch 
1  square  decimeter  =    .1076  of  a  square  foot 
1  square  meter  =  1.196  square  yards 

1  are  =  3.954  square  rods 

1  hektare  =  2.471  acres 

1  square  kilometer  =    .3861  of  a  square  mile 

Cubic  Measure 

1  cubic  inch  =  16.387  cubic  centimeters 
1  cubic  foot  =  28.317  cubic  decimeters 
1  cubic  yard  =      .7645  of  a  cubic  meter 
1  cord  =    3.624  steres 

1  cubic  centimeter  :=    .061  of  a  cubic  inch 
1  cubic  decimeter    =    .0353  of  a  cubic  foot 
1  cubic  meter  =  1.308  cubic  yards 

1  stere  =    .2759  of  a  cord 


TABLES  317 

Measures  of  Capacity 

1  liquid  quart  =    .9463  of  a  liter 
1  dry  quart       =  1.101  liters 
1  liquid  gallon  =    .3785  of  a  dekaliter 
1  peck  =    .881  of  a  dekaliter 

1  bushel  =    .3524  of  a  hektoliter 

1  liter  =  1.0567  liquid  quarts 

1  liter  =    .908  of  a  dry  quart 

1  dekaliter  =  2.6417  liquid  gallons 
1  dekaliter  =1.135  pecks 
1  hektoliter  =  2.8375  bushels 

Measures  of  Weight 

1  grain  Troy  =      .0648  of  a  gram 

1  ounce  Avoirdupois  =  28.35  grams 

1  ounce  Troy  =  31.104  grams 

1  pound  Avoirdupois  =     .4536  of  a  kilogram 

1  pound  Troy  =      .3732  of  a  kilogram 

1  ton  (short)  =      .9072  of  a  tonneau 

1  gram        =     .03527  of  an  ounce  Avoirdupois 

1  gram        =      .03215  of  an  ounce  Troy 

1  gram        =  15.432  grains  Troy 

1  kilogram  =    2.2046  pounds  Avoirdupois 

1  kilogram  =    2.679  pounds  Troy 

1  tonneau  =    1.1023  tons  (short) 

CONVENIENT  MULTIPLES   FOR  CONVERSION 

To  Convert 

Grains  to  Grams,  multiply  by         .066 

Ounces  to  Grams,                        multiply  by  28.35 

Pounds  to  Grams,                       multiply  by  453.6 

Pounds  to  Kilograms,                 multiply  by  .46 

Hundredweights  to  Kilograms,  multiply  by  50.8 
Tons  to  Kilograms,                     multiply  by  1016. 

Grams  to  Grains,                        multiply  by  15.4 

Grams  to  Ounces,                        multiply  by  .36 

Kilograms  to  Ounces,                  multiply  by  35.3 

Kilograms  to  Pounds,                  multiply  by  2.2 

Kilograms  to  Hundredweights,  multiply  by  .02 


318  MATHEMATICAL  WEINKLES 


Kilograms  to  Tons, 

multiply  by- 

.001 

Inches  to  Millimeters, 

multiply  by 

25.4 

Inches  to  Centimeters, 

multiply  by 

2.54 

Feet  to  Meters, 

multiply  by 

.3048 

Yards  to  Meters, 

multiply  by 

.9144 

Yards  to  Kilometers, 

multiply  by 

.0009 

Miles  to  Kilometers, 

multiply  by 

1.6 

Millimeters  to  Inches, 

multiply  by 

.04 

Centimeters  to  Inches, 

multiply  by 

.4 

Meters  to  Feet, 

multiply  by 

3.3 

Meters  to  Yards, 

multiply  by 

1.1 

Kilometers  to  Yards, 

multiply  by  1093.6 

Kilometers  to  Miles, 

multiply  by 

.62 

MISCELLANEOUS 

Acre  =  5645.376  square  varas. 

Acre  (square)  is  209f  feet  each  way. 

Ampere  (unit  of  current)  is  that  current  of  electricity  that  decom- 
poses .00009324  gram  of  water  per  second. 

Are  =  a  square  dekameter. 

Barleycorn  =  ^  inch. 

Barrel  (flour)  weighs  196  pounds. 

Barrel  (wine)  holds  31  gallons. 

Bushel  (imperial)  =  2216.192  cubic  inches. 

Bushel  (U.  S.)  =  2150.42  cubic  inches. 

Cable  length  =  120  fathoms. 

Calorie  =  42,000,000  ergs  =  .428  kilogrammeter. 

Carat  (assayer's  weight)  =:  10  pennyweight. 

Carat  (of  diamond)  =  3^  grains.  , 

Centare  =  1  square  meter. 

Century  =  100  years. 

Chaldron  =  36  bushels. 

Coulomb  (unit  of  quantity)  is  a  current  of  1  ampere  during  1  second 
of  time. 

Crown  =  5  shillings. 

Cubic  foot  of  water  weighs  62|  pounds. 

Cubit  =  18  inches. 

Cycle  (metonic)  =  19  years. 

Cycle  (of  indiction)  =  15  years. 

Cycle  (solar)  =28  years. 


TABLES  319 

Degree  (1°)  =  ^  of  a  right  angle  =  ■—-  radian. 

180 

Dozen  =  12. 

Dozen  (baker's)  =  13. 

Eagle  =  a  10. 

Farthing  =  S  .00503. 

Fathom  =  6  feet. 

Firkin  (wine  measure)  =  9  gallons. 

Florin  (Austrian)  =  84.53. 

Fortnight  =  2  weeks. 

Furlong  =  J  mile. 

Gallon  (drj')  =  268  cubic  inches. 

Gallon  (liquid)  =  231  cubic  inches. 

Gram  =  weight  of  1  cubic  centimeter  of  distilled  water  at  its  maximum 
density. 

Great  gross  =  12  gross. 

Gross  =  12  dozen. 

Gross  ton,  long  ton  =  2240  pounds. 

Guilder  (Holland)  =  S.402. 

Guinea  =  21  shillings. 

Half  section  =  320  acres. 

Hand  =  4  inches. 

Heat  of  fusion  of  ice  at  0°  C.  is  80  calories  per  gram. 

Heat  of  vaporization  of  water  at  100°  C.  is  536  calories  per  gram. 

Hectare  =  1  square  hectometer. 

Hogshead  =  2  barrels. 

Kilo  =  a  kilogram. 

Knot  =  6086  feet,  or  1.15  miles. 

Labor  =  177.136  acres. 

League  or  Sitio  (Spanish)  =  4428.4  acres. 

Leap  year.    The  centennial  years  divisible  by  400  and  all  other  years 
divisible  by  4  are  leap  years. 

Light  travels  300,000,000  meters,  or  180,000  miles,  per  second. 

Liter  =  1  cubic  decimeter. 

Long  hundredweight  =112  pounds. 

Mill  =  8 .001. 

Minim  =  a  drop  of  pure  water. 

Mite  =  f  .0187. 

Nautical  mile  =  1  knot. 

Ohm   (unit  of  resistance)  is  the  electrical  resistance  of  a  column  of 
mercury  106  centimeters  long  and  of  1  square  millimeter  section. 


320  MATHEMATICAL   WEINKLES 

Pace  (common)  =  3  feet. 

Pace  (military)  =  2^  feet. 

Pack  =  240  pounds. 

Parcian  (Spanish)  =  5314.08  acres. 

Penny  =$.02025. 

Period  (Dionysian,  or  Paschal)  =  532  years. 

Quarter  (English)  =  8  bushels;  U.  S.  =  8^  bushels. 

Quarter  section  =  160  acres. 

Quintal  =  100,000  grams. 

Radian  =  —  =  57.2957796°. 

IT 

Eod  =  5|  yards,  or  16^  feet. 

Score  =  20. 

Section  =  640  acres. 

Sextant  =  60°. 

Shilling  =  1 .243. 

Sign  =  30  degrees. 

Span  =  9  inches. 

Specific  heat  of  ice  is  about  0.506. 

Square  =  100  square  feet. . 

Stere  =  .2759  cord,  or  1  cubic  meter. 

Stone  =  14  pounds. 

Strike  (dry  measure)  =  2  bushels. 

Ton  (long)  =  2240  pounds. 

Ton  (register)  =  100  cubic  feet. 

Ton  (shipping)  =  40  cubic  feet. 

Ton  (short)  =  2000  pounds. 

Tonneau  =  1.1023  tons. 

Township  =  36  square  miles. 

Vara  (California)  =  33  inches. 

Vara  (Texas)  =  .9259+  yard,  or  33^  inches. 

Volt  (unit  of  electromotive  force)  is  1  ampere  of  current  passing 
through  a  substance  having  1  ohm  of  resistance. 

Watt  (unit  of  power)  is  the  power  of  1  ampere  current  passing  through 
a  resistance  of  1  ohm. 

Year  (common)  =  365  days. 

Year  (leap,  or  bissextile)  =  366  days. 

Year  (lunar)  =  354  days. 

Year  (sidereal)  =  365  days,  6  hours,  9  minutes,  9  seconds. 

Year  (solar)  =  365  days,  5  hours,  48  minutes,  46.05  seconds. 


INDEX 


Age  Table 

Algebraic  Problems 

Answers  and  Solutions  to 

Algebraic  Problems     .     .     .     . 

Arithmetical  Problems    .     .    . 

Greometrical  Exercises     .    .    . 

Mathematical  Recreations  .    . 

Miscellaneous  Problems  .    .     . 

Approximate  Results 

Arithmetical  Problems  .     .     .     . 

Arithmetical  Series 

Belts 

Bins,  Cisterns,  etc 

Brick  and  Stone  Work    .     .    .    . 

Carpeting 

Casks  and  Barrels 

Compound  Interest  Tables  .  .  . 
Cube  Roots  of  Integers  .     .     .     . 

Density  of  a  Body 

Examination  Questions  .  .  . 
Extraction  of  Any  Root .     .     .     . 

Foarth  Dimension 

Fractions  Classified 

Geometrical  Exercises  .  .  .  . 
Geometrical  Magnitudes  Classi- 
fied       

Grain  and  Hay 

G.  C.  D.  of  Fractions .     .     .     .     . 

Harmonic  Mean 

Historical  Notes 

Historical  Notes  on  Arithmetic  . 

Homer's  Methotl 

Interest     

L.  C.  M.  of  Fractions      .     .     .     . 

I^ogs 

Lumber 

Marking  Goods 

Mathematical  Branches  Defined 
Mathematical  Recreations  .     .    , 

Mathematical  Signs 

Mathematics  Classified  .  .  .  , 
Mean  Proportional 


75 
25 

185 
163 

im 

205 
201 
240 
1 
286 
258 
259 
259 
260 
260 
305 
307 
265 
113 
289 
108 
302 
33 

303 
267 
288 
287 
294 
296 
289 
237 
288 
»)8 


244 
291 
58 
301 
302 
287 


Mensuration 258 

Metric  Tables 314 

Miscellaneous  Helps 285 

Miscellaneous  Problems      ...  48 

Multiplication  Table 304 

Nine  Point  Circle 45 

Numbers  Classified 302 

Painting  and  Plastering      .     .     .  269 

Papering 270 

Periods  of  Numeration   ....  286 

Pi  (it)  to  707  places 285 

Quotations  on  Mathematics    .    .  245 
Right  Triangles  whose  Sides  are 

Integral 306 

Roofing  and  Flooring     ....  274 
Scalene  Triangles  whose  Areas 

are  Integral 306 

Scientific  Truths 291 

Short  Methods 228 

Addition 228 

Approximate  Results  ....  240 

Division 234 

Fractions 235 

Multiplication 230 

Interest 237 

Subtraction 230 

Similar  Solids 276 

Similar  Surfaces 276 

Specific  Gravities 309 

Squares  of  Integers 306 

Square  Roots  of  Integers     .     .     .  307 

Table  of  Equivalents 316 

Table  of  Prime  Numbers    ...  308 

Tables 304 

To  find  the  Day  of  the  Week  .    .  287 
To  find  the  Day's  Length  at  Any 

Longitude 288 

To  find  the  Height  of  a  Stump    .  289 

To  Sura  to  Infinity 287 

Values  of  Foreign  Coins               .  310 

Weights  and  Measures    ....  311 

Wood  Measure 283 


321 


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